2010 Today's Calculation Of Integral

Part of Today's Calculation Of Integral

Subcontests

(148)

1

Today's calculation of Integral 669

Find the differentiable function defined in

x>0

{\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).

1

Today's calculation of Integral 668

Consider two curves

y=\sin x,\ y=\sin 2x

0\leq x\leq 2\pi

(\alpha ,\ \beta)\ (0<\alpha <\pi)

be the intersection point of the curves. If

\sin x-\sin 2x

has a local minimum at

x=x_1

and a local maximum at

x=x_2

, then find the values of

\cos x_1,\ \cos x_1\cos x_2

.
(2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of

\alpha \leq x\leq \pi

for the figure about the line

y=-1

.
2011 Kyorin　University entrance exam/Medicine

1

Today's calculation of Integral 667

a>1,\ 0\leq x\leq \frac{\pi}{4}

. Find the volume of the solid generated by a rotation of the part bounded by two curves

y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}

x

-axis.
1993 Hiroshima Un iversity entrance exam/Science

1

Today's calculation of Integral 666

f(x)

be a function defined in

0<x<\frac{\pi}{2}

satisfying:
(i)

f\left(\frac{\pi}{6}\right)=0

f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt

f(x)

.
1987 Sapporo Medical University entrance exam

1

Today's calculation of Integral 665

\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)

.
1992 Japan Women's University entrance exam/Physics, Mathematics

1

Today's calculation of Integral 66４

For a positive integer

n

I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx

I_1+I_2+I_3+I_4

.
1992 University of Fukui entrance exam/Medicine

1

Today's calculation of Integral 663

Given are the curve

y=x^2+x-2

and a curve which is obtained by tranfering the curve symmetric with respect to the point

(p,\ 2p)

p

change in such a way that these two curves intersects, find the maximum area of the part bounded by these curves. 1978 Nagasaki University entrance exam/Economics

1

Today's calculation of Integral 662

xyz

A

be the solid generated by a rotation of the figure, enclosed by the curve

y=2-2x^2

x

-axis about the

y

-axis.
(1) When the solid is cut by the plane

x=a\ (|a|\leq 1)

, find the inequality which expresses the figure of the cross-section.
(2) Denote by

L

the distance between the point

(a,\ 0,\ 0)

and the point on the perimeter of the cross-section found in (1), find the maximum value of

L

.
(3) Find the volume of the solid by a rotation of the solid

A

x

-axis.
1987 Sophia University entrance exam/Science and Technology

1

Today's calculation of Integral 661

Consider a sequence

1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots

.
(1) Find the 36th term.
(2) Find

\int x^2\ln x\ dx

A

be the product of from the first term to the 36th term. How many digits does

A

have integer part?
If necessary, you may use the fact

2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31

.
2010 National Defense Medical College Entrance Exam, Problem 4

1

Today's calculation of Integral 660

a,\ b

be given positive constants.
Evaluate

\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.

1

Today's calculation of Integral 659

\int_0^1 \frac{\ln (x+2)}{x+1}dx.

1

Today's calculation of Integral 658

Consider a parameterized curve

C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).

(1) Find the length

L

C

.
(2) Find the area

S

of the region enclosed by the

x,\ y

C

.
Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School.
1997 Kyoto University entrance exam/Science

1

Today's calculation of Integral 657

a_n

\int_{a_n}^{a_{n+1}} (1+|\sin x|)dx=(n+1)^2\ (n=1,\ 2,\ \cdots),\ a_1=0

\lim_{n\to\infty} \frac{a_n}{n^3}

1

Today's calculation of Integral 656

\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).

1

Today's calculation of Integral 655

Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas

y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)

in such a way that there existed the points of tangency in the first quadrant.

1

Today's calculation of Integral 654

f(x)

x\geq 0

\lim_{x\to\infty} \frac{f(x)}{x}=1

\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx

.
1997 Hokkaido University entrance exam/Science

1

Today's calculation of Integral 652

a,\ b,\ c

be positive real numbers such that

b^2>ac.

\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.

1981 Tokyo University, Master Course

1

Today's calculation of Integral 651

\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.

1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology

1

Today's calculation of Integral 650

Find the values of

p,\ q,\ r\ (-1<p<q<r<1)

such that for any polynomials with degree

\leq 2

, the following equation holds:

\int_{-1}^p f(x)\ dx-\int_p^q f(x)\ dx+\int_q^r f(x)\ dx-\int_r^1 f(x)\ dx=0.

1995 Hitotsubashi University entrance exam/Law, Economics etc.

1

Today's calculation of Integral 649

f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})

,

I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).

\lim_{n\to\infty} I_n.

2009 Tokyo Institute of Technology, Master Course in Mathematics

1

Today's calculation of Integral 648

Consider a function real-valued function with

C^{\infty}

\mathbb{R}

f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.

x\neq 0,\ f(x)>0.

Judge whether the following integrals

(i),\ (ii)

converge or diverge, justify your answer.

(i)

\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.

(ii)

\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.

2010 Kyoto University, Master Course in Mathematics

1

Today's calculation of Integral 647

\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})

1

Today's calculation of Integral 646

\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})

1

Today's calculation of Integral 645

Prove the following inequality.

\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}

1

Today's calculation of Integral 644

p

\int_1^p e^xdx=1

\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.

1

Today's calculation of Integral 643

\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.

1

Today's calculation of Integral 642

\int_0^{\frac{\pi}{6}} \frac{(\tan ^ 2 2x)\sqrt{\cos 2x}+2}{(\cos ^ 2 x)\sqrt{\cos 2x}}dx.

1

Today's calculation of Integral 641

\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.

1

Today's calculation of Integral 640

\int_0^{\frac{\pi}{4}} \frac{1}{1-\sin x}\sqrt{\frac{\cos x}{1+\cos x+\sin x}}dx.

1

Today's calculation of Integral 639

\int_0^1 (x+3)\sqrt{xe^x}\ dx.

1

Today's calculation of Integral 638

(a,\ b)

be a point on the curve

y=\frac{x}{1+x}\ (x\geq 0).

U

the volume of the figure enclosed by the curve , the

x

axis and the line

x=a

, revolved around the the

x

axis and denote

V

the volume of the figure enclosed by the curve , the

y

axis and th line

y=b

, revolved around the

y

axis. What's the relation of

U

V?

1978 Chuo university entrance exam/Science and Technology

1

Today's calculation of Integral 637

For a non negative integer

n

I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx

to answer the following questions:
(1) Calculate

I_{n+2}+I_n.

(2) Evaluate the values of

I_1,\ I_2

I_3.

1978 Niigata university entrance exam

1

Today's calculation of Integral 636

a>1

be a constant. In the

xy

A(a,\ 0),\ B(a,\ \ln a)

C

be the intersection point of the curve

y=\ln x

x

-axis. Denote by

S_1

the area of the part bounded by the

x

-axis, the segment

BA

y=\ln x

1\leq b\leq a

D(b,\ \ln b)

. Find the value of

b

such that the area of quadrilateral

ABDC

is the closest to

S_1

and find the area

S_2

\lim_{a\rightarrow \infty} \frac{S_2}{S_1}

.
1992 Tokyo University entrance exam/Science

1

Today's calculation of Integral 635

Suppose that a function

f(x)

-1<x<1

satisfies the following properties (i) , (ii), (iii).
(i)

f'(x)

is continuous.
(ii) When

-1<x<0,\ f'(x)<0,\ f'(0)=0

0<x<1,\ f'(x)>0

f(0)=-1

F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)

F(\sin \theta)=c\theta\ (c :\text{constant})

-\frac{\pi}{2}<\theta <\frac{\pi}{2}

f(x)

.
1975 Waseda University entrance exam/Science and Technology

1

Today's calculation of Integral 634

\int_1^{\sqrt{e}} (\ln x)^n dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^m\ (n=1,\ 2,\ \cdots)

2010 Miyazaki University entrance exam/Medicine

1

Today's calculation of Integral 633

f(x)

be a differentiable function. Find the value of

x

\{f(x)\}^2+(e+1)f(x)+1+e^2-2\int_0^x f(t)dt-2f(x)\int_0^x f(t)dt+2\left\{\int_0^x f(t)dt\right\}^2

is minimized.
1978 Tokyo Medical College entrance exam

1

Today's calculation of Integral 632

\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).

2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam

1

Today's calculation of Integral 631

\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}

Proposed by kunny

1

Today's calculation of Integral 630

\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.

1

Today's calculation of Integral 629

\int_0^{\infty} \frac{1}{e^{x}(1+e^{4x})}dx.

1

Today's calculation of Integral 628

(1) Evaluate the following definite integrals.
(a)

\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx

\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx

\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx

a

be a positive constant. Find the area of the cross section cut by the plane

z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)

of the solid such that

x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0

, then find the volume of the solid.
1984 Yamanashi Medical University entrance exam
Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them.
Thanks in advance.
kunny

1

Today's calculation of Integral 627

\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(2\sin \theta +1)\cos ^ 3 \theta}{(\sin ^ 2 \theta +1)^2}d\theta .

Proposed by kunny

1

Today's calculation of Integral 626

\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.

2010 Nara Medical University entrance exam

1

Today's calculation of Integral 625

\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).

2010 Kumamoto University entrance exam/Medicine

1

Today's calculation of Integral 624

Find the continuous function

f(x)

such that the following equation holds for any real number

x

.

\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.

1977 Keio University entrance exam/Medicine

1

Today's calculation of Integral 623

Find the continuous function satisfying the following equation.

\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.

1978 Shibaura Institute of Technology entrance exam

1

Today's calculation of Integral 622

0<k<2

, consider two curves

C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).

S(k)

the sum of the areas of four parts enclosed by

C_1,\ C_2

x=0,\ x=\pi

. Find the minimum value of

S(k).

2010 Nagoya Institute of Technology entrance exam

1

Today's calculation of Integral 621

\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).

2010 Yokohama National University entrance exam/Engineering, 2nd exam

1

Today's calculation of Interal 620

a,\ b

be real numbers. Suppose that a function

f(x)

f(x)=a\sin x+b\cos x+\int_{-\pi}^{\pi} f(t)\cos t\ dt

and has the maximum value

2\pi

-\pi \leq x\leq \pi

. Find the minimum value of

\int_{-\pi}^{\pi} \{f(x)\}^2dx.

2010 Chiba University entrance exam

1

Today's calculation of Integral 619

Consider a function

f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).

a

be the value of

x

f(x)

is maximized.
Evaluate

\int_a^{\frac{\pi}{2}} f(x)\ dx.

2010 Saitama University entrance exam/Mathematics
Last Edited

1

Today's calculation of Integral 618

Find the minimu value of

\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \{x\cos t+(1-x)\sin t\}^2dt.

2010 Ibaraki University entrance exam/Science

1

Today's calculation of Integral 617

y=f(x)

be a function of the graph of broken line connected by points

(-1,\ 0),\ (0,\ 1),\ (1,\ 4)

x

y

plane. Find the minimum value of

\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.

2010 Tohoku University entrance exam/Economics, 2nd exam

1

Today's calculation of Integral 616

\int_1^3 \frac{\ln (x+1)}{x^2}dx

.
2010 Hirosaki University entrance exam

1

Today's calculation of Integral 615

0\leq a\leq 2

, find the minimum value of

\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.

2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.

1

Today's calculation of Integral 614

\int_0^1 \{x(1-x)\}^{\frac 32}dx.

2010 Hirosaki University School of Medicine entrance exam

1

Today's calculation of Integral 613

Find the area of the part, in the

x

y

plane, enclosed by the curve

|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).

2010 Tokyo University of Agriculture and Technology entrance exam

1

Today's calculation of Integral 612

f(x)=\frac{1}{x}\ (x>0)

, prove the following inequality.

f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}

1

Today's calculation of Integral 611

g(t)

be the minimum value of

f(x)=x2^{-x}

t\leq x\leq t+1

\int_0^2 g(t)dt

.
2010 Kumamoto University entrance exam/Science

1

Today's calculation of Integral 610

\int_2^a \frac{x^a-1-xa^x\ln a}{(x^a-1)^2}dx.

proposed by kunny

1

Today's calculation of Integral 609

Prove that for positive number

t

F(t)=\int_0^t \frac{\sin x}{1+x^2}dx

always takes positive number.
1972 Tokyo University of Education entrance exam

1

Today's calculation of Integral 608

a>0

, find the minimum value of

\int_0^1 \frac{ax^2+(a^2+2a)x+2a^2-2a+4}{(x+a)(x+2)}dx.

2010 Gakusyuin University entrance exam/Science

1

Today's calculation of Integral 607

On the coordinate plane, Let

C

be the graph of

y=(\ln x)^2\ (x>0)

\alpha >0

L(\alpha)

be the tangent line of

C

(\alpha ,\ (\ln \alpha)^2).

(1) Draw the graph.
(2) Let

n(\alpha)

be the number of the intersection points of

C

L(\alpha)

n(\alpha)

0<\alpha <1

S(\alpha)

be the area of the region bounded by

C,\ L(\alpha)

x

S(\alpha)

.
2010 Tokyo Institute of Technology entrance exam, Second Exam.

1

Today's calculation of Integral 606

Find the area of the part bounded by two curves

y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1

x

-axis.
1956 Tokyo Institute of Technology entrance exam

1

Today's calculation of Integral 605

f(x)

be a differentiable function. Find the following limit value:

\lim_{n\to\infty} \dbinom{n}{k}\left\{f\left(\frac{x}{n}\right)-f(0)\right\}^k.

Especially, for

f(x)=(x-\alpha)(x-\beta)

find the limit value above.
1956 Tokyo Institute of Technology entrance exam

1

Today's calculation of Integral 604

r

be a positive integer. Determine the value of

a

for which the limit value

\lim_{n\to\infty} \frac{\sum_{k=1}^n k^r}{n^a}

has a non zero finite value, then find the limit value.
1956 Tokyo Institute of Technology entrance exam

1

Today's calculation of Integral 603

Find the minimum value of

\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx

.
Please solve the problem without using partial differentiation for those who don't learn it.
1961 Waseda University entrance exam/Science and Technology

1

Today's calculation of Integral 602

Prove the following inequality.

\frac{e-1}{n+1}\leqq\int^e_1(\log x)^n dx\leqq\frac{(n+1)e+1}{(n+1)(n+2)}\ (n=1,2,\cdot\cdot\cdot)

1994 Kyoto University entrance exam/Science

1

Today's calculation of Integral 601

\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx

.
created by kunny

1

Today's calculation of Integral 600

\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)

.
created by kunny

1

Today's calculation of Integral 599

\int_0^{\frac{\pi}{6}} \frac{e^x(\sin x+\cos x+\cos 3x)}{\cos^ 2 {2x}}\ dx

.
created by kunny

1

Today's calculation of Integral 598

a

C(a)

x\geq 1

y=\sqrt{x^2-1}+\frac{a}{x}

.
(1) Find the maximum value

a_0

a

C(a)

is contained to lower part of

y=x

y<x

0<\theta <\frac{\pi}{2}

, find the volume

V(\theta)

V

obtained by revoloving the figure bounded by

C(a_0)

and three lines

y=x,\ x=1,\ x=\frac{1}{\cos \theta}

x

-axis.
(3) Find

\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)

.
1992 Tokyo University entrance exam/Science, 2nd exam

1

Today's calculation of Integral 597

In space given a board shaped the equilateral triangle

PQR

P\left(1,\ \frac 12,\ 0\right),\ Q\left(1,-\frac 12,\ 0\right),\ R\left(\frac 14,\ 0,\ \frac{\sqrt{3}}{4}\right)

S

is revolved about the

z

-axis, find the volume of the solid generated by the whole points through which

S

passes.
1984 Tokyo University entrance exam/Science

1

Today's calculation of Integral 596

Find the minimum value of

\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).

2009 Shimane University entrance exam/Medicine

1

Today's calculation of Integral 595

\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.

2009 Kumamoto University entrance exam/Medicine

1

Today's calculation of Integral 594

x

y

plane, two variable points

P,\ Q

P(2t,\ -2t^2+2t),\ Q(t+2,-3t+2)

t

t_0

as the time such that

\overline{PQ}=0

t

varies in the range of

0\leq t\leq t_0

, find the area of the region swept by the line segment

PQ

x

y

1

Today's calculation of Integral 593

For a positive integer

m

, prove the following ineqaulity.
0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}
1996 Osaka University entrance exam

1

Today's calculation of Integral 592

Prove the following inequality. \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})

1

Today's calculation of Integral 591

a,\ b,\ c

be real numbers such that

a\geq b\geq c\geq 1

. Prove the following inequality: \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.

1

Today's calculation of Integral 590

Evaluate \int_0^{\frac{\pi}{8}} \frac{(\cos \theta \plus{}\sin \theta)^{\frac{3}{2}}\minus{}(\cos \theta \minus{}\sin \theta)^{\frac{3}{2}}}{\sqrt{\cos 2\theta}}\ d\theta.

1

Today's calculation of Integral 589

Evaluate \int_0^1 \frac{x}{\{(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\plus{}(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx.

1

Today's calculation of Integral 588

Evaluate \int_0^{\frac{\pi}{2}} e^{xe^x}\{(x\plus{}1)e^x(\cos x\plus{}\sin x)\plus{}\cos x\minus{}\sin x\}dx.

1

Today's calculation of Integral 587

Evaluate \int_0^1 \frac{(x^2\plus{}3x)e^x\minus{}(x^2\minus{}3x)e^{\minus{}x}\plus{}2}{\sqrt{1\plus{}x(e^x\plus{}e^{\minus{}x})}}\ dx.

1

Today's calculation of Integral 586

Evaluate \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx.

1

Today's calculation of Integral 585

Evaluate \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx.

1

Today's calculation of Integral 584

Find \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right).

1

Today's calculation of Integral 583

Find the values of

k

such that the areas of the three parts bounded by the graph of y\equal{}\minus{}x^4\plus{}2x^2 and the line y\equal{}k are all equal.

1

Today's calculation of Integral 582

Prove the following inequality. \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi

1

Today's calculation of Integral 581

c

for which cx^2\geq \ln (1\plus{}x^2) for all real numbers

x

, find the value of

c

such that the area of the figure bounded by two curves y\equal{}cx^2 and y\equal{}\ln (1\plus{}x^2) and two lines x\equal{}1,\ x\equal{}\minus{}1 is 4.

1

Today's calculation of Integral 522

Find \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx.

1

Today's calculation of Integral 580

k

be a positive constant number. Denote

\alpha ,\ \beta \ (0<\beta <\alpha)

x

coordinates of the curve C: y\equal{}kx^2\ (x\geq 0) and two lines l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}.　Find the minimum area of the part bounded by the curve

C

l,\ m

1

Today's calculation of Integral 579

a

be a positive real number. Find \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}

1

Today's calculation of Integral 578

Find the range of

k

for which the following inequality holds for

0\leq x\leq 1

. \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}} If necessary, you may use \ln 3 \equal{} 1.10.

1

Today's calculation of Integral 577

Prove the following inequality for any integer

N\geq 4

. \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5

1

Today's calculation of Integral 576

For a function f(x)\equal{}(\ln x)^2\plus{}2\ln x, let

C

be the curve y\equal{}f(x). Denote

A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)

the points of tangency of two tangents drawn from the origin

O

C

C

. Answer the following questions. (1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve

C

. (2) Find the values of

a,\ b

. (3) Find the volume by a rotation of the figure bounded by the part from the point

A

B

and line segments

OA,\ OB

y

1

Today's calculation of Integral 575

For a function f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right), answer the following questions. (1) Find

f'(x)

n

th term of the sequence

a_n

such that a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots).

1

Today's calculation of Integral 574

n

be a positive integer. Prove that x^ne^{1\minus{}x}\leq n! for

x\geq 0

1

Today's calculation of Integral 573

Find the area of the figure bounded by three curves C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right) C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right) C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right).

1

Today's calculation of Integral 572

n,\ a_n

is difined by a_n\equal{}\int_0^{\frac{\pi}{4}} (\cos x)^ndx. (1) Find a_{\minus{}2},\ a_{\minus{}1}. (2) Find the relation of

a_n

and a_{n\minus{}2}. (3) Prove that a_{2n}\equal{}b_n\plus{}\pi c_n for some rational number

b_n,\ c_n

c_n

n<0

1

Today's calculation of Integral 571

Evaluate \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx.

1

Today's calculation of Integral 570

Let f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x. (1) Show that f(x) \equal{} 0 has a unique solution in

0 < x < \pi

. (2) Let J \equal{} \int_0^{\pi} |f(x)|dx. Denote by

\alpha

the solution in (1), express

J

\sin \alpha

. (3) Compare the size of

J

defined in (2) with

\sqrt {2}

1

Today's calculation of Integral 569

In the coordinate plane, denote by

S(a)

the area of the region bounded by the line passing through the point

(1,\ 2)

a

and the parabola y\equal{}x^2. When

a

varies in the range of

0\leq a\leq 6

, find the value of

a

S(a)

1

Today's calculation of Integral 568

n

2n

boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let

p_n

be the probability such that any box has ball less than or equal to one. Find the limit

\lim_{n\to\infty} \frac{\ln p_n}{n}

1

Today's calculation of Integral 567

a

be a positive real numbers. In the coordinate plane denote by

S

the area of the figure bounded by the curve

y=\sin x\ (0\leq x\leq \pi)

x

-axis and denote

T

by the area of the figure bounded by the curves y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\　y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right) and the

x

-axis. Find the value of

a

S: T=3: 1

1

Today's calculation of Integral 566

In the coordinate space, consider the cubic with vertices

O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)

. Find the volume of the solid generated by revolution of the cubic around the diagonal

OF

as the axis of rotation.

1

Today's calculation of Integral 565

Prove that f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt has maximal value at x\equal{}\frac 12.

1

Today's calculation of Integral 564

In the coordinate plane with

O(0,\ 0)

, consider the function C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2} and two distinct points

P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)

C

. (1) Let H_i\ (i \equal{} 1,\ 2) be the intersection points of the line passing through P_i\ (i \equal{} 1,\ 2), parallel to

x

axis and the line y \equal{} x. Show that the area of

\triangle{OP_1H_1}

\triangle{OP_2H_2}

are equal. (2) Let

x_1 < x_2

. Express the area of the figure bounded by the part of

x_1\leq x\leq x_2

C

and line segments

P_1O,\ P_2O

y_1,\ y_2

1

Today's calculation of Integral 563

Determine the pair of constant numbers

a,\ b,\ c

such that for a quadratic function f(x) \equal{} x^2 \plus{} ax \plus{} b, the following equation is identity with respect to

x

. f(x \plus{} 1) \equal{} c\int_0^1 (3x^2 \plus{} 4xt)f'(t)dt .

1

Today's calculation of Integral 562

(1) Show the following inequality for every natural number

k

. \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k} (2) Show the following inequality for every natural number

m,\ n

m > n

. \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}

1

Today's calculation of Integral 561

Evaluate \int_{\minus{}1}^1 \frac{1\plus{}2x^2\plus{}3x^4\plus{}4x^6\plus{}5x^8\plus{}6x^{10}\plus{}7x^{12}}{\sqrt{(1\plus{}x^2)(1\plus{}x^4)(1\plus{}x^6)}}dx.

1

Today's calculation of Integral 560

K

be the figure bounded by the graph of function y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}},

x

axis and the line x \equal{} \frac {1}{2}. (1) Find the volume

V_1

of the solid generated by rotation of

K

x

axis. (2) Find the volume

V_2

of the solid generated by rotation of

K

y

axis. Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.

1

Today's calculation of Integral 559

xyz

space, consider two points P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0). Let

S

be the surface generated by rotation the line segment

PQ

x

axis. Answer the following questions. (1) Find the volume of the solid bounded by the surface

S

and two planes x\equal{}1 and x\equal{}\minus{}1. (2) Find the cross-section of the solid in (1) by the plane y\equal{}0 to sketch the figure on the palne y\equal{}0. (3) Evaluate the definite integral \int_0^1 \sqrt{t^2\plus{}1}\ dt by substitution t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}. Then use this to find the area of (2).

1

Today's calculation of Integral 558

For a positive constant

t

\alpha ,\ \beta

be the roots of the quadratic equation x^2 \plus{} t^2x \minus{} 2t \equal{} 0. Find the minimum value of \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.

1

Today's calculation of Integral 557

Find the folllowing limit. \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).

1

Today's calculation of Integral 556

Prove the following inequality.

\sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}

Last Edited. Sorry, I have changed the problem. kunny

1

Today's calculation of Integral 555

\frac {1}{e} < t < 1

, find the minimum value of \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx.

1

Today's calculation of Integral 554

Use \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1}) to evaluate \int_0^1 \sqrt{4x^2\plus{}1}dx.

1

Today's calculation of Integral 553

Find the continuous function such that f(x)\equal{}\frac{e^{2x}}{2(e\minus{}1)}\int_0^1 e^{\minus{}y}f(y)dy\plus{}\int_0^{\frac 12} f(y)dy\plus{}\int_0^{\frac 12}\sin ^ 2(\pi y)dy.

1

Today's calculation of Integral 552

Find the positive value of

a

such that the curve C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}} tangent to the parabola C_2: y \equal{} ax^2, then find the equation of the tangent line of

C_1

at the point of tangency.

1

Today's calculation of Integral 551

In the coordinate plane, find the area of the region bounded by the curve C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1} and the line L: y\equal{}1.

1

Today's calculation of Integral 550

Evaluate \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}.

1

Today's calculation of Integral 549

f(x)

be a function defined on

[0,\ 1]

n=1,\ 2,\ 3,\ \cdots

P_n(x)

P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}

\lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx

1

Today's calculation of Integral 548

For f(x)\equal{}e^{\frac{x}{2}}\cos \frac{x}{2}, evaluate \sum_{n\equal{}0}^{\infty} \int_{\minus{}\pi}^{\pi}f(x)f(x\minus{}2n\pi)dx\ (n\equal{}0,\ 1,\ 2,\ \cdots).

1

Today's calculation of Integral 547

Find the minimum value of \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty).

1

Today's calculation of Integral 546

Find the minimum value of \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx.

1

Today's calculation of Integral 545

\int_0^1 xe^{x^2}dx

. (2) Let I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx. Express I_{n\plus{}1} in terms of

I_n

1

Today's calculation of Integral 544

(1) Evaluate \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx.
(2) If a linear function

f(x)

satifies \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)f(x)dx\equal{}3\sqrt{3}, then we have f(x)\equal{}\boxed{\ A\ }(x\minus{}1)\plus{}\boxed{\ B\ }(x\plus{}1), thus we have f(x)\equal{}\boxed{\ C\ }.

1

Today's calculation of Integral 543

y

be the function of

x

satisfying the differential equation y'' \minus{} y \equal{} 2\sin x. (1) Let y \equal{} e^xu \minus{} \sin x, find the differential equation with which the function

u

with respect to

x

satisfies. (2) If y(0) \equal{} 3,\ y'(0) \equal{} 0, then determine

y

1

Today's calculation of Integral 542

Find continuous functions

f(x),\ g(x)

which takes positive value for any real number

x

, satisfying g(x)\equal{}\int_0^x f(t)\ dt and \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1.

1

Today's calculation of Integral 541

Find the functions

f(x),\ g(x)

satisfying the following equations. (1) f'(x) \equal{} 2f(x) \plus{} 10,\ f(0) \equal{} 0 (2) \int_0^x u^3g(u)du \equal{} x^4 \plus{} g(x)

1

Today's calculation of Integral 540

Evaluate \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx.

1

Today's calculation of Integral 539

\int_0^{\frac{\pi}{4}} \frac{\sin ^ 2 x}{\cos ^ 3 x}\ dx

1

Today's calculation of Integral 538

Evaluate \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx.

1

Today's calculation of Integral 537

Evaluate \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx.

1

Today's calculation of Integral 536

Evaluate \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx.

1

Today's calculation of Integral 535

C

be the parameterized curve for a given positive number

r

0\leq t\leq \pi

, C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \\ y \equal{} 2r\sin ^ 2 t & \end{array} \right. When the point

P

moves on the curve

C

, (1) Find the magnitude of acceleralation of the point

P

t

. (2) Find the length of the locus by which the point

P

0\leq t\leq \pi

. (3) Find the volume of the solid by rotation of the region bounded by the curve

C

x

-axis about the

x

1

Today's calculation of Integral 534

Find the indefinite integral \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx.

1

Today's calculation of Integral 533

C

be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length

2\pi

A(1,\ 0)

and put the other end point

P

P_0(1,\ 2\pi)

. From this situation, when we twist the string around

C

by moving the point

P

in anti clockwise with the string streched tightly, find the length of the curve that the point

P

draws from sarting point

P_0

to reaching point

A

1

Today's calculation of Integral 532

For a curve C: y \equal{} x\sqrt {9 \minus{} x^2}\ (x\geq 0), (1) Find the maximum value of the function. (2) Find the area of the figure bounded by the curve

C

x

-axis. (3) Find the volume of the solid by revolution of the figure in (2) around the

y

-axis. Please find the volume without using cylindrical shells for my students. Last Edited.

1

Today's calculation of Integral 531

f(x)

be a continuous function defined on

[a,\ b]

, it is known that there exists some

c

such that \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b) Explain the fact by using graph. Note that you don't need to prove the statement. (2) Let f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n, Prove that there exists

\theta

such that f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.

1

Today's calculation of Integral 530

Answer the following questions.
(1) By setting x\plus{}\sqrt{x^2\minus{}1}\equal{}t, find the indefinite integral \int \sqrt{x^2\minus{}1}\ dx.
(2) Given two points

P(p,\ q)\ (p>1,\ q>0)

A(1,\ 0)

on the curve x^2\minus{}y^2\equal{}1. Find the area

S

of the figure bounded by two lines

OA,\ OP

and the curve in terms of

p

.
(3) Let S\equal{}\frac{\theta}{2}. Express

p,\ q

\theta

1

Today's calculation of Integral 529

Prove that the following inequality holds for each natural number

n

. \int_0^{\frac {\pi}{2}} \sum_{k \equal{} 1}^n \left(\frac {\sin kx}{k}\right)^2dx < \frac {61}{144}\pi

1

Today's calculation of Integral 528

Consider a function f(x)\equal{}xe^{\minus{}x^3} defined on any real numbers. (1) Examine the variation and convexity of

f(x)

to draw the garph of

f(x)

. (2) For a positive number

C

D_1

be the region bounded by y\equal{}f(x), the

x

-axis and x\equal{}C. Denote

V_1(C)

the volume obtained by rotation of

D_1

x

\lim_{C\rightarrow \infty} V_1(C)

M

be the maximum value of y\equal{}f(x) for

x\geq 0

D_2

the region bounded by y\equal{}f(x), the

y

-axis and y\equal{}M. Find the volume

V_2

obtained by rotation of

D_2

y

1

Today's calculation of Integral 527

n,\ m

be positive integers and

\alpha ,\ \beta

be real numbers. Prove the following equations. (1) \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3 (2) \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2} (3) \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}

1

Today's calculation of integral 526

For a function satisfying

f'(x) > 0

a\leq x\leq b

, let F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt. For what value of

x

F(x)

1

Today's calculation of Integral 525

a,\ b

be real numbers satisfying \int_0^1 (ax\plus{}b)^2dx\equal{}1. Determine the values of

a,\ b

for which \int_0^1 3x(ax\plus{}b)\ dx is maximized.

1

Today's calculation of Integral 524

Evaluate the following definite integral. 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}

1

Today's calculation of Integral 523

Prove the following inequality. \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}

1

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