2008 Today's Calculation Of Integral

Part of Today's Calculation Of Integral

Subcontests

(138)

1

Today's calculation of Integral 395

3 points O(0,\ 0),\ P(a,\ a^2), Q( \minus{} b,\ b^2)\ (a > 0,\ b > 0) are on the parabpla y \equal{} x^2. Let

S_1

be the area bounded by the line

PQ

and the parabola and let

S_2

be the area of the triangle

OPQ

. Find the minimum value of

\frac {S_1}{S_2}

1

Today's calculation of Integral 394

Find \lim_{x\rightarrow 0} \frac{1}{x} \int_{\minus{}x}^x (t\sin 2006 t\plus{}2007t\plus{}1004)\ dt.

1

Today's calculation of Integral 393

V(a)

be the volume of the solid obtained by revolvong the region bounded by the curve y \equal{} x\sqrt {x\sin ax}\ (0\leq x\leq \frac {\pi}{a}) and

x

axis around the

x

a > 0

, find the minimum value of V(a) \plus{} V\left(\frac{1}{a}\right).

1

Today's calculation of Integral 392

Evaluate \int_{ \minus{} 1}^1 \frac {|x|}{1 \plus{} e^x}\ dx.

1

Today's calculation of Integral 391

Evaluate \int_0^{\frac {\pi}{2}} \sin |2x \minus{} a|\ dx for a real number

a\in{[0,\ \pi]}

1

Today's calculation of Integral 390

Find the polynomials

f(x),\ g(x)

such that: \frac{1}{\pi}\int_0^{\frac{1}{2}} \frac{tf'(t)\minus{}6g(t)}{\sqrt{1\minus{}t^2}}\ dt\equal{}f(x)\minus{}g(x)\plus{}x \frac{6}{\pi}\int_0^{\frac{1}{2}} \frac{8f(t)\minus{}5g'(t)}{\sqrt{1\minus{}t^2}}\ dt\equal{}2f(x)\minus{}3g(x)\minus{}x^2\plus{}2x.

1

Today's calculation of Integral 389

Find the values of

t\in{[0,\ 2]}

for which \int_{t\minus{}3}^{2t} 2^{x^2}\ dx is maximal and minimal.

1

Today's calculation of Integral 388

For f(x)\equal{}\sin ^ 3 x, let

p(x)

be quadratic polynomial. Evaluate

\int_0^{2\pi} p(x)f''(x)\ dx

1

Today's calculation of Integral 387

l_1,\ l_2

be the tangent and nomal line respectively at the point (p,\ \ln (p \plus{} 1)) on the curve C: y \equal{} \ln (x \plus{} 1). Denote by T_i\ (i \equal{} 1,\ 2) the areas bounded by l_i\ (i \equal{} 1,\ 2), C and the

y

axis respectively. Find the limit

\lim_{p\rightarrow 0} \frac {T_2}{T_1}

1

Today's calculation of Integral 386

For a\equal{}2007*2009, evaluate \int_0^1 \frac{6x\plus{}5a\sqrt{x}}{4\sqrt{x\plus{}a\sqrt{x}}}\ dx.

1

Today's calculation of Integral 385

Evaluate \int_0^{\frac{\pi ^2}{4}} (2\sin \sqrt{x}\plus{}\sqrt{x}\cos \sqrt{x})\ dx.

1

Today's calculation of Integral 384

Evaluate \int_\frac{\pi}{4}^{\frac {\pi}{3}} (\sin x \plus{} \cos x)\sqrt {\frac {e^x}{\sin x}}\ dx.

1

Today's calculation of Integral 383

For a positive integer

m

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

1

Today's calculation of Integral 382

a>0,\ b\geq 0

, Compare \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}. (2) Find \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}.

1

Today's calculation of Integral 381

f(x)

is defined as follows for

x\geq 0

. f(x) \equal{} \sin \left(\frac {n\pi}{4}\right)\sin x\ (n\pi \leq x < (n \plus{} 1)\pi)\ (n \equal{} 0,\ 1,\ 2,\ \cdots). Evaluate

\int_0^{100\pi} f(x)\ dx

1

Today's calculation of Integral 380

\int_{ - 1}^1 {n\choose k}(1 + x)^{n - k}(1 - x)^k\ dx\ (k = 0,\ 1,\ 2,\ \cdots n)

1

Today's calculation of Integral 379

\alpha ,\ r

be real numbers such that

r > 1,\ r\neq 3,\ r\neq 4

. Find the values of

\alpha ,\ r

such that \lim_{n\to\infty} \sum_{k \equal{} 1}^n \frac {n^{\alpha}}{(n \plus{} k)^{r}} \equal{} \frac {r \minus{} 3}{(r \minus{} 1)(r \minus{} 4)}.

1

Today's calculation of Integral 378

Evaluate \int_0^{\pi} x|\sin nx|\ dx\ (n\equal{}1,\ 2,\ \cdots).

1

Today's calculation of Integral 377

The line y\equal{}mx has 3 intersection points with the curve y\equal{}|x(x\minus{}1)|. Find the value of

m

such that the area of the 2 regions bounded by the line and the curve are equal.

1

Today's calculation of Integral 376

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

1

Today's calculation of Integral 375

Prove the following inequality.

\frac {1}{n} < \int_0^{\frac {\pi}{2}} \frac {1}{(1 + \cos x)^n}\ dx < \frac {n + 5}{n(n + 1)}\ (n =2,3,\ \cdots)

1

Today's calculation of Integral 374

n\geq 2

be positive integers. (1) Prove that n\ln n \minus{} n\plus{}1<\sum_{k\equal{}1}^n \ln k<(n\plus{}1)\ln n \minus{} n\plus{}1. (2) Find

\lim_{n\to\infty} (n!)^{\frac{1}{n\ln n}}

1

Today's calculation of Integral 373

Evaluate \int_0^{\frac{\pi}{2}} e^x\left\{\cos \ (\sin x)\cos ^2 \frac{x}{2}\plus{}\sin \ (\sin x)\sin ^ 2 \frac{x}{2}\right\}\ dx.

1

Today's calculation of Integral 372

Evaluate
\displaystyle\int^1_0 e^{x \plus{} e^{x \plus{} e^{x \plus{} e^x}}}\ dx.

1

Today's calculation of Integral 371

\int_{ - 1}^1 {\frac {1}{(1 + e^x)(1 + x^2)} \ dx. }

1

Today's calculation of Integral 370

Calculate \int_0^{\pi/3} \frac{\cos^{2}x \plus{} 1}{\cos x \sqrt {\cos x}}\ dx.

1

Today's calculation of Integral 369

\displaystyle\int\limits^{e^{e^{e^e}}}_{e^{e^e}} \dfrac{1}{xlnxln(lnx)ln(ln(lnx))}dx

1

Today's calculation of Integral 368

For a real number

a

, evaluate \int_0^a |x(x \minus{} a^2)|\ dx.

1

Today's calculation of Integral 367

Let f(x)\equal{}x^2\plus{}\int_0^1 xf(t)\ dt\plus{}\int_{\minus{}1}^2 f(t)\ dt. Evaluate \int_0^{\frac{\pi}{2}} f(x\plus{}\sin x)\ dx.

1

Today's calculation of Integral 366

(1) Determine the constant numbers

a,\ b,\ c,\ p,\ q,\ r,\ s

such that the following equation is equality. \ \ \ \ \ 4x \equal{} \{a(x \plus{} 1)^2 \plus{} b(x \plus{} 1) \plus{} c\}(x^2 \plus{} 1)^2 \plus{} \{(px \plus{} q)(x^2 \plus{} 1) \plus{} (rx \plus{} s)\}(x \plus{} 1)^3. (2) Evaluate the following definite integrals. (a) \displaystyle \int_0^1 \frac {dx}{x \plus{} 1}\ \ \ \ \ \ \ \ \ (b)\ \int_0^1 \frac {dx}{(x \plus{} 1)^2}\ \ \ \ \ (c)\ \int_0^1 \frac {dx}{(x \plus{} 1)^3} (d) \displaystyle \int_0^1 \frac {x}{x^2 \plus{} 1}\ dx\ \ \ \ (e)\ \int_0^1 \frac {dx}{x^2 \plus{} 1}\ \ \ \ \ \ \ (f)\ \int_0^1 \frac {x}{(x^2 \plus{} 1)^2}\ dx (g) \displaystyle \int_0^1 \frac {dx}{(x^2 \plus{} 1)^2}\ \ \ \ (h)\ \int_0^1 \frac {4x}{(x \plus{} 1)^3(x^2 \plus{} 1)^2}\ dx

1

Today's calculation of Integral 365

a,\ b

be postive constant numbers. Evaluate \int_0^{\frac {\pi}{2}} \frac {a\cos x \minus{} b\sin x}{b\sin x \plus{} a\cos x}\ dx.

1

Today's calculation of Integral 364

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

1

Today's calculation of Integral 363

t\geq 0

, let a_n\equal{}\int_0^ t e^{nx}\ dx\ (n\equal{}0,\ 1,\ 2,\ 3). (1) Show that a_3\minus{}3a_2\plus{}3a_1\minus{}a_0\geq 0. (2) Show that e^ta_0\plus{}(e^t\minus{}1)a_1\minus{}a_2\geq 0.

1

Today's calculation of Integral 362

Evaluate the following definite integrals. (1)

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

. (2) \int_1^2 \frac{1}{x(x\plus{}1)}\ dx. (3) \int_0^1 \frac{1}{1\plus{}e^x}\ dx. (4)

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

1

Today's calculation of Integral 361

Find the following indefinite integrals. (1) \int \frac{\sin x}{3\plus{}\sin ^ 2 x}\ dx. (2) \int e^{2x\plus{}e^x}\ dx.

1

Today's calculation of Integral 360

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

1

Today's calculation of Integral 359

Evaluate \int_{\minus{}1}^1 \frac{2x^4\minus{}x^3\minus{}2x^2\plus{}1}{x^3\minus{}x\plus{}\sqrt{1\minus{}x^2}}\ dx.

1

Today's calculation of Integral 358

a

be a postive constant number. Given a positive integer

n

, take an integer

m

such that m\leq \frac{na}{\pi}

1

Today's calculation of Integral 357

0<a<1

S(a)

is the area of the figure bounded by three curves y\equal{}e^x,\ y\equal{}e^{\frac{1\plus{}a}{1\minus{}a}x} and y\equal{}e^{2\minus{}x}. Find

\lim_{a\rightarrow 0} \frac{S(a)}{a}

1

Today's calculation of Integral 356

A continuous function

f(x)

satisfies that \frac {d}{dx}\left(\int_0^x f(x \plus{} t)\ dt \right)\equal{} 0. Find the necessary and sufficient condition such that \sum_{n \equal{} 1}^{\infty} f(2^n) \equal{} 1.

1

Today's calculation of Integral 355

For a positive number

n

, find \lim_{n\to\infty} n^2\int_{e^{\minus{}\frac{1}{n}}}^1 x^n \ln x\ dx.

1

Today's calculation of Integral 354

Evaluate \int_{\minus{}2}^2 x^4\sqrt{4\minus{}x^2}\ dx.

1

Today's calculation of Integral 353

Consider a parabola C: y\equal{}\frac{1}{4}x^2 and the point

F(0,\ 1)

. For the origin

O

n

points on the parabola

C

A_1(x_1,\ y_1),\ A_2(x_2,\ y_2) ,\ \cdots \ , A_n(x_n,\ y_n)

x_k>0

and \angle{OFA_k}\equal{}\frac{k\pi}{2n}\ (k\equal{}1,\ 2,\ 3,\ \cdots\ , n). Find \lim_{n\to\infty} \frac{1}{n} \sum_{k\equal{}1}^n FA_k.

1

Today's calculation of Integral 352

Prove the following inequality. 1\leq \lim_{n\to\infty} \int_{\frac{1}{n}}^{\frac{\pi}{2}} \frac{\pi \sin x}{x(\pi \minus{}x)}\ dx<3.

1

Today's calculation of Integral 351

Find the positive value of

k

for which \int_0^{\frac {\pi}{2}} |\cos x \minus{} kx|\ dx is minimized.

1

Today's calculation of Integral 350

Evaluate \int_0^1 (1\minus{}x^2)^{\frac{5}{2}} dx.

1

Today's calculation of Integral 349

For real numbers

a,\ b\ (ab\neq 0)

, evaluate \int_0^{\pi} (a\sin x\plus{}b\cos x)^6 dx.

1

Today's calculation of Integral 348

Find \lim_{n\to\infty} \left(\frac{n\plus{}1}{n}\right)^{\frac{1}{n^2}}\left(\frac{n\plus{}2}{n}\right)^{\frac{2}{n^2}}\left(\frac{n\plus{}3}{n}\right)^{\frac{3}{n^2}}\cdots \left(\frac{n\plus{}n}{n}\right)^{\frac{n}{n^2}}\ (n\equal{}1,\ 2,\ 3,\ \cdots).

1

Today's calculation of Integral 347

For real positive numbers

a

, find the minimum value of \int_0^{\pi} \left(\frac{x}{a}\plus{}a\sin x\right)^2 dx.

1

Today's calculation of Integral 346

Suppose that two curves C_1 : y\equal{}x^3\minus{}x, C_2 : y\equal{}(x\minus{}a)^3\minus{}(x\minus{}a) have two intersection points. Find the maximum area of the figure bounded by

C_1,\ C_2

1

Today's calculation of Integral 345

Given a continuous function

f(x)

such that \int_0^{2\pi} f(x)\ dx \equal{} 0. Let S(x) \equal{} A_0 \plus{} A_1\cos x \plus{} B_1\sin x, find constant numbers

A_0,\ A_1

B_1

for which \int_0^{2\pi} \{f(x) \minus{} S(x)\}^2\ dx is minimized.

1

Today's calculation of Integral 343

Find all continuous positive functions

f(x)

0\leq x\leq 1

, such that \int_0^1 f(x)\ dx \equal{} 1 \int_0^1 xf(x)\ dx \equal{} \alpha \int_0^1 x^2f(x)\ dx \equal{} \alpha ^2 where

\alpha

is given real number.

1

Today's calculation of Integral 342

Prove the following inequality. \left(\int_0^1 (x^2\plus{}px\plus{}q)\ dx\right)^2\leq \int_0^1 (x^2\plus{}px\plus{}q)^2\ dx\minus{}\frac{1}{180}

1

Today's calculation of Integral 341

c

be a constant number. Find the real value of

x

such that \int_0^c |t\minus{}x|\ dt\equal{}\int_0^x |t\minus{}c|\ dt.

1

Today's calculation of Integral 340

Find the continuous function

f(x)

such that xf(x)\minus{}\int_0^x f(t)\ dt\equal{}x\plus{}\ln (\sqrt{x^2\plus{}1}\minus{}x) with f(0)\equal{}\ln 2.

1

Today's calculation of Integral 339

Find the minimum area of the part bounded by the parabola y\equal{}a^3x^2\minus{}a^4x\ (a>0) and the line y\equal{}x.

1

Today's calculation of Integral 338

Given a parameterized curve C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}. Find the area bounded by the curve

C

x

axis and two lines x\equal{}\pm 1.

1

Today's calculation of Integral 337

Find \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \sum_{k\equal{}1}^n \sin kx \sin x\ dx.

1

Today's calculation of Integral 336

Let P_n\equal{}\sqrt[n]{\frac{(3n)!}{(2n)!}}\ (n\equal{}1,\ 2,\ \cdots). Find the following limits. (1)

\lim_{n\to\infty} \frac{P_n}{n}

. (2) \lim_{n\to\infty} \left(\frac{n\plus{}2}{n}\right)^{P_n}.

1

Today's calculation of Integral 335

t>0

\frac{1}{2t}\int_t^{3t} x\ln x\ dx\geq 2t\ln 2t

1

Today's calculation of Integral 334

Evaluate \int_0^1 \frac{7x^3\plus{}23x^2\plus{}21x\plus{}15}{(x^2\plus{}1)(x\plus{}1)^2}\ dx.

1

Today's calculation of Integral 333

Find the functions

f(x),\ g(x)

such that f(x)\plus{}\int_0^x g(t)\ dt\equal{}\sin x(\cos x\minus{}\sin x),\ \{f'(x)\}^2\plus{}\{g(x)\}^2\equal{}1.

1

Today's calculation of Integral 332

f(x)

be a function such that 1\minus{}f(x)\equal{}f(1\minus{}x) for

0\leq x\leq 1

\int_0^1 f(x)\ dx

1

Today's calculation of Integral 331

a\in\mathbb{R}

, find the minimum value of \int_0^{\frac {\pi}{2}} \left|\frac {\sin 2x}{1 \plus{} \sin ^ 2 x} \minus{} a\cos x\right| dx.

1

Today's calculation of Integral 330

Find all real numbers

x

such that \int_0^x t^2\sin (x\minus{}t)\ dt\equal{}x^2.

1

Today's calculation of Integral 329

f(x),\ g(x)

be continuous functions defined in the range of

0\leq x\leq 1

f(x)

such that f(x)\equal{}\int_0^1 e^{x\plus{}t}f(t)\ dt. (2) Find

g(x)

such that g(x)\equal{}\int_0^1 e^{x\plus{}t}g(t)\ dt\plus{}x.

1

Today's calculation of Integral 328

t

be a negative real number and

D

be the part bounded by the curve y\equal{}2^{2x\plus{}2t}, the curve y\equal{}2^{x\plus{}3t} and the

y

axis. (1) Find the volume

V(t)

generated by rotation of

D

x

t

moves in the range of negative real number, find the maximum value of

V(t)

1

Today's calculation of Integral 327

a,\ b

be real numbers and

C

be the graph of the function y\equal{}e^{a\plus{}bx^2}. (1) Find the values of

a,\ b

C

passes through the point

P(1,\ 1)

and the slope of the tangent line of

C

P

is \minus{}2. (2) For the values of

a,\ b

(1)

, find the volume of the solid generated by the revolution of the part which lies in the right side for the

y

axis in the figure bounded by the parabola y\equal{}x^2 and the curve

C

y

1

Today's calculation of Integral 326

Evaluate \int_1^e \frac{(x\plus{}1)(x\plus{}2\minus{}\ln x)}{(x\plus{}\ln x)^2}\ dx.

1

Today's calculation of Integral 325

Prove that \ \frac {1}{3} < \int_0^1 x^{(\sin x \plus{} \cos x)^2}\ dx < \frac {1}{2}.

1

Today's calculation of Integral 324

Evaluate \int_1^e \left(\frac {1}{\sqrt {x\ln x}} \plus{} \sqrt {\frac {\ln x}{x}}\right)\ dx.

1

Today's calculation of Integral 323

Evaluate \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left(\sqrt{\frac{\sin x}{x}}\plus{}\sqrt{\frac{x}{\sin x}}\cos x\right)\ dx.

1

Today's calculation of Integral 322

Evaluate \int_1^e \frac {(x^4 \minus{} 4x^3 \plus{} 9x^2 \minus{} 6x \plus{} 8)e^x}{x^3}\ dx.

1

Today's calculation of Integral 321

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

1

Today's calculation of Integral 320

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

1

Today's calculation of Integral 319

Evaluate \int_{\minus{}1}^1 \frac{xe^x\plus{}e^x\plus{}1}{x^2(e^x\plus{}1)^2}\ dx.

1

Today's calculation of Integral 318

Evaluate \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1\plus{}\cot x}{e^x\sin x}\ dx.

1

Today's calculation of Integral 317

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

1

Today's calculation of Integral 316

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

1

Today's calculation of Integral 315

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

1

Today's calculation of Integral 314

Evaluate \int_{\frac{\pi}{2}}^{\pi} x^{\sin x}(1\plus{}x\cos x\cdot\ln x \plus{}\sin x)\ dx.

1

Today's calculation of Integral 313

Find the area bounded by the graph of y\equal{}\sqrt[3]{x\plus{}\sqrt{x^2\plus{}1}}\plus{}\sqrt[3]{x\minus{}\sqrt{x^2\plus{}1}}, the line x\minus{}y\minus{}1\equal{}0 and the

x

1

Today's calculation of Integral 312

a,\ b

be postive real numbers. For a real number

t

d(t)

the distance between the origin and the line (ae^t)x \plus{} (be^{ \minus{} t})y \equal{} 1.
Let

a,\ b

vary with ab \equal{} 1, find the minimum value of

\int_0^1 \frac {1}{d(t)^2}\ dt

1

Today's calculation of Integral 311

Prove the following inequality. \int_0^1 2008^{x^{2008}}\ dx\geq \left(1\minus{}\frac{1}{2008^{2008}}\right)\frac{1}{\ln 2008}.

1

Today's calculation of Integral 310

Define the function

f(x)

as : \displaystyle f(x)\equal{}\left\{ \begin{array}{ll} \frac{\ln (1\plus{}x)}{1\plus{}x}\ (x\geq 0) & \\ \frac{\ln (1\minus{}x)}{1\minus{}x}\ (\minus{}1\leq x<0) & \end{array} \right. 1. Examine the variation of

f(x)

and find the maximum and minimum value of

f(x)

. 2. Find the value of

a

for which \int_{\minus{}1}^{e\minus{}1} |f(x)\minus{}a|\ dx is minimized for

0\leq a\leq \frac{1}{e}

1

Today's calculation of Integral 309

(1) Calculate the indefinite integral : \int e^{\minus{}x}\sin 2x\ dx. (2) Evaluate the definite integral : \int_0^{\pi} e^{\minus{}x}|\sin 2x|\ dx.

1

Today's calculation of Integral 308

a

be a positive constant number. For a positive integer

n

, define a function

I_n(t)

by I_n(t)\equal{}\int_0^t x^ne^{\minus{}ax}dx. Answer the following questions. Note that you may use \lim_{t\rightarrow \infty} t^ne^{\minus{}at}\equal{}0 without proof. (1) Evaluate

I_1(t)

. (2) Find the relation of I_{n\plus{}1}(t),\ I_n(t). (3) Prove that there exists

\lim_{t\to\infty} I_n(t)

for all natural number

n

by using mathematical induction. (4) Find

\lim_{t\to\infty} I_n(t)

1

Today's calculation of Integral 307

For real numbers

a,\ b,\ c

, let f(x) \equal{} ax^2 \plus{} bx \plus{} c. Prove that : \int_{ \minus{} 1}^1 (1 \minus{} x^2)\{f'(x)\}^2 dx\leq 6\int_{ \minus{} 1}^1 \{f(x)\}^2 dx.

1

Today's calculation of Integral 306

For positive real numbers

a,\ b

, two graphs of the function :

x^a

\ln bx

have a tangency of point. (1) Let

(s,\ t)

be the tangency of point, express this in terms of

a

b

as the function of

a

h

0 < h < s

, denote the area as

A(h)

of the domain bounded by the line x \equal{} h and two curves y \equal{} x^a,\ y \equal{} \ln bx. Express

\lim_{h\rightarrow 0} A(h)

a

1

Today's calculation of Integral 305

Find \int_0^{\infty} (1\minus{}\sqrt{1\minus{}e^{\minus{}2x}})\ dx.

1

Today's calculation of Integral 304

\alpha ,\ \beta

be real numbers with

0\leq \alpha \leq \beta

and f(x)\equal{}x^2\minus{}(\alpha \plus{}\beta)x\plus{}\alpha \beta such that \int_{\minus{}1}^1 f(x)\ dx\equal{}1. Find the maximum value of

\int_0^{\alpha} f(x)\ dx

1

Today's calculation of Integral 303

In the x \minus{} y plane, find the area of the region bounded by the parameterized curve as follows. \left\{\begin{array}{ll} x \equal{} \cos 2t & \\ y \equal{} t\sin t & \end{array} \right.\ (0\leq t\leq 2\pi)

1

Today's calculation of Integral 302

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

1

Today's calculation of Integral 301

For the positive constant number

a

D

be the part surrounded by the curve \sqrt{x}\plus{}\sqrt{y}\equal{}\sqrt{a} and the line x\plus{}y\equal{}a. (1) Draw the outline of

D

and find the area. (2) Find the volume of the solid by rotating

D

about the line x\plus{}y\equal{}a as the rotation axis.

1

Today's calculation of Integral 300

In Euclidean space, take the point

N(0,\ 0,\ 1)

S

with radius 1 centered in the origin. For moving points

P,\ Q

S

such that NP \equal{} NQ and \angle{PNQ} \equal{} \theta \left(0 < \theta < \frac {\pi}{2}\right), consider the solid figure

T

in which the line segment

PQ

can be passed. (1) Show that

z

P,\ Q

are equal. (2) When

P

is on the palne z \equal{} h, express the length of

PQ

\theta

h

. (3) Draw the outline of the cross section by cutting

T

by the plane z \equal{} h, then express the area in terms of

\theta

h

. (4) Pay attention to the range for which

h

can be valued, express the volume

V

T

\theta

, then find the maximum

V

\theta

1

Today's calculation of Integral 299

Let I_n(x)\equal{}\int_1^x (\ln t)^ndt\ (x>0) for n\equal{}1,\ 2,\ 3,\ \cdots. (1) Prove by mathematical induction that

I_n(x)

is expressed by I_n(x)\equal{}xf_n(\ln x)\plus{}C_n\ (n\geq 1) in terms of some polynomial

f_n(y)

n

and some constant number

C_n.

(2) Express the constant term of

f_n(y)

n.

1

Today's calculation of Integral 298

A

be the region : \left\{(x,\ y)|x\geq 0,\ y\geq 0, \left(\frac{x}{a}\right)^{\frac{1}{2}}\plus{}\left(\frac{y}{1\minus{}a}\right)^{\frac{1}{2}}\leq 1\right\} for

0<a<1.

(1) Find the area of

A.

(2) Find the maximum volume of the solid obtained by revolving

A

x

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Today's calculation of Interal 297

\int_0^{\frac{\pi}{6}} \left(\sin \frac{3}{2}x\right)\left(\cos \frac{x}{2}\right)\ dx.

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Today's calculation of Integral 296

Let a_n\equal{}\int_0^{\frac{\pi}{2} } (1\minus{}\sin t)^n\sin 2t\ dt. (1) Find \sum_{n\equal{}1}^{\infty} a_n. (2) Find \sum_{n\equal{}1}^{\infty} \frac{a_n}{n}. (3) Find \sum_{n\equal{}1}^{\infty} (n\plus{}1)(a_n\minus{}a_{n\plus{}1}).

1

Today's calculation of Integral 295

Let f_n(x) \equal{} x^n(1 \minus{} x)^n,\ I_n \equal{} \int_0^1 f_n(x)\ dx\ (n \equal{} 1,2,\cdots). (1) Find a polynomial

g(x)

such that f_{n \plus{} 1}'(x) \equal{} (n \plus{} 1)f_n(x)g(x). (2) Find constant numbers

A_n,\ B_n

such that f_{n \plus{} 2}''(x) \equal{} A_nf_{n \plus{} 1}(x) \plus{} B_nf_n(x). (3) Find A_nI_{n \plus{} 1} \plus{} B_nI_n. (4) Let J_n \equal{} (2n \plus{} 1)!I_n. Express J_{n \plus{} 1} in terms of

J_n

I_n

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Today's calculation of Integral 294

Evaluate \int_{\ln 2}^{\ln 3} \frac {\ln (e^x \plus{} 1)}{e^{2x}}\ dx.

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Today's calculation of Integral 293

Consider the parabolas C_a: y \equal{} \frac {1}{4}x^2 \minus{} ax \plus{} a^2 \plus{} a \minus{} 2 and C: y \equal{} \minus{} \frac {1}{4}x^2 \plus{} 2 for real number

a

in the x \minus{} y plane. (1) Find the equation of the locus of the vertex for

C_a.

(2) For a \equal{} 3, find the slope of the two common tangent lines of

C

C_a,

then the intersection points of the lines. (3) Suppose that

C_a

intersects with

C

at two distinct points. Find the maximum area of the figure bounded by

C

C_a.

1

Today's calculation of Integral 292

x,\ y

t

be real numbers such that

\{ \begin{array}{ll} t^3 - (2y - 1)t^2 - t + 2y - 1 = 0 & &emsp; \\ xt - 2t + 2y = 0 & &emsp; \end{array}

x - y

plane find the area of the figure bounded by the part of curves above with

x\geq 0

y = 2.

1

Today's calculation of Integral 291

Consider the parabola C: y \equal{} x^2 in the x \minus{} y plane. Let

l_1

be the tangent line of

C

\sqrt {3}

l_2

be the tangent line of

C

with negative slope which makes angle

45^\circ

l_1

. Find the area bounded by

C,\ l_1

l_2

1

Today's calculation of Integral 290

Let the curve C: y \equal{} |x^2 \plus{} 2x \minus{} 3| and the line

l

passing through the point ( \minus{} 3,\ 0) with a slope

m

in x\minus{}y plane. Suppose that

C

l

at distinct two points other than the point ( \minus{} 3,\ 0), find the value of

m

for which the area of the figure bounded by

C

l

1

Today's calculation of Integral 289

p

be a positive constant number, find

\lim_{a\to\infty} \int_0^p |x\sin (ax^2)|\ dx.

1

Today's calculation of Integral 288

Evaluate \int_0^1 (1\plus{}2008x^{2008})e^{x^{2008}}\ dx.

1

Today's calculation of Integral 287

For constant numbers

a,\ b

, let f(x)\equal{}e^{\minus{}x}(ax\plus{}b). Suppose that the curve y\equal{}f(x) passes through two points (t,\ 1),\ (t\plus{}1,\ 1). Find the area of the part bounded by the curve y\equal{}f(x) and the line y\equal{}1.

1

Today's calculation of Integral 286

Evaluate \int_{\minus{}2008}^{2008} \frac{f'(x)\plus{}f'(\minus{}x)}{2008^x\plus{}1}\ dx

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Today's calculation of Integral 285

Find the minimum value of \int_0^1 \frac{2}{|2t\minus{}x|\plus{}3}\ dt　for

0\leq x\leq 4.

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Today's calculation of Integral 284

f(x)

is a continuous function which takes positive values for

x\geq 0.

f(x)

such that \int_0^x f(t)dt\equal{}x\sqrt{f(x)} with f(1)\equal{}\frac{1}{2}.

1

Today's calculation of Integral 283

f(x)

is a continuous function with the piriodicity of

2\pi

c

is a positive constant number. Find

f(x)

c

such that \int_0^{2\pi} f(t \minus{} x)\sin tdt \equal{} cf(x) with f(0) \equal{} 1 for all real numbers

x

1

Today's calculation of Integral 282

g(x)

is a differentiable function for

0\leq x\leq \pi

g'(x)

is a continuous function for

0\leq x\leq \pi

. Let f(x) \equal{} g(x)\sin x. Find

g(x)

such that \int_0^{\pi} \{f(x)\}^2dx \equal{} \int_0^{\pi}\{f'(x)\}^2dx.

1

Today's calculation of Integral 280

p,\ q\ (p < q)

x

coordinates of curves x^2 \plus{} y^2 \equal{} 1,\ y\geq 0 and y \equal{} \frac {1}{4x}. (1) Find

\alpha ,\ \beta

such that \cos \frac {\alpha}{2} \equal{} p,\ \cos \frac {\beta }{2} \equal{} q\ (0 < \alpha < \pi ,\ 0 < \beta < \pi). (2) Find the area bounded by the curves.

1

Today's calculation of Integral 281

Find the maximum and minimum values of \int_0^{\pi} (a\sin x\plus{}b\cos x)^3dx for

|a|\leq 1,\ |b|\leq 1.

1

Today's calculation of Integral 278

Find the value of

t

{ \frac{\int_0^{\frac{\pi}{2}}(\sin x+t\cos x)dx}{\sqrt{\int_0^{\frac{\pi}{2}} (\sin x+t\cos x)^2dx}}}

1

Today's calculation of Integral 277

Find the fuction such that f(x)\equal{}x^2\minus{}x\left(2\int_0^1 |f(t)|\ dt\right)^{\frac{1}{2}}.

1

Today's calculation of Integral 279

If the function f(x) \equal{} 3x^2 \plus{} 2ax \plus{} b satisfies \int_{ \minus{} 1}^1 |f(x)|\ dx < 2, then show that f(x) \equal{} 0 has distinct real roots.

1

Today's calculation of Integral 276

Prove that \int_{\frac{1}{n}}^{\pi \minus{}\frac{1}{n}} \frac{1}{\sqrt[3]{\sin x}}\ dx\leq \frac{3}{2}\pi \ (n\equal{}1,\ 2,\ 3,\ \cdots).

1

Today's calculation of Integral 275

The line y \equal{} mx has three intersection points with the curve y \equal{} |x(x \minus{} 1)|. Find the value of

m

such that the areas of two parts bounded by the line and the curve are equal.

1

Today's calculation of Integral 274

For real constant numbers

a,\ b,\ c,\ d,

consider the function f(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d such that f( \minus{} 1) \equal{} 0,\ f(1) \equal{} 0,\ f(x)\geq 1 \minus{} |x| for

|x|\leq 1.

f(x)

for which \int_{ \minus{} 1}^1 \{f'(x) \minus{} x\}^2\ dx　is minimized.

1

Today's calculation of Integral 273

Find the area bounded by 0\leq y\leq \frac{x|\sin x|}{1\plus{}|\cos x|},\ 0\leq x\leq n\pi \ (n\equal{}1,\ 2,\ \cdots).

1

Today's calculation of Integral 272

\lim_{n\to\infty} \int_0^{\pi} e^x|\sin nx|dx.

1

Today's calculation of Integral 271

For a positive constant number

a

f(x)

satisfies the following equation. f(x)\equal{}xe^{\minus{}\frac{x}{a}}\plus{}\frac{1}{a\plus{}1}\int_0^a f(t) dt (1) Express the value of

\int_0^a f(t)dt

a.

(2) Find the maximum and minimum value of

f(x)

for \minus{}1\leq x\leq 1.

1

Today's calculation of Integral 270

f(x)

be the continuous function at

0\leq x\leq 1

such that \int_0^1 x^kf(x)\ dx\equal{}0 for integers k\equal{}0,\ 1,\ \cdots ,\ n\minus{}1\ (n\geq 1). (1) For all real numbers

t,

find the minimum value of g(t)\equal{}\int_0^1 |x\minus{}t|^n\ dx. (2) Show the following equation for all real real numbers

t.

\int_0^1 (x\minus{}t)^nf(x)\ dx\equal{}\int_0^1 x^nf(x)\ dx (3) Let

M

be the maximum value of the function

|f(x)|

0\leq x\leq 1.

Show that \left|\int_0^1 x^nf(x)\ dx\right|\leq \frac{M}{2^n(n\plus{}1)}

1

Today's calculation of Integral 269

C: y = \frac {1}{1 + x^2}

A(\alpha ,\ f(\alpha)),\ B\left( - \frac {1}{\alpha},\ f\left( - \frac {1}{\alpha} \right)\right)\ (\alpha > 0).

Find the minimum area bounded by the line segments

OA,\ OB

C,

O

is the origin.
Note that you are not allowed to use the integral formula of

\frac {1}{1 + x^2}

for the problem.

1

Today's calculation of Integral 268

Find the constant numbers

u,\ v,\ s,\ t\ (s<t)

such that \int_{\minus{}1}^1 f(x)\ dx\equal{}uf(s)\plus{}vf(t) holds for any polynomials f(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d with the degree of 3 or less than.

1

Today's calculation of Integral 267

K

be the curved surface obtained by rotating the parabola y \equal{} \frac {3}{4} \minus{} x^2 about the

y

K

H

passing through the origin and forming angle

45^\circ

for the axis. Find the volume of the solid surrounded by

K

H.

Note that you are not allowed to use Double Integral for the problem.

1

Today's calculation of Integral 266

Find the area of the region expressed by the system of inequality \frac{x^2}{4}\plus{}y^2\leq 1,\ \left(1\plus{}\frac{\sqrt{3}}{2}\right)x\plus{}y\leq 1.

1

Today's calculation of Integral 265

f(x)

f'(x)

and for any real numbers

x,\ y

, \int_y^{x \plus{} y} f(t)\ dt \equal{} \int_0^x \{f(y)\cos t \plus{} f(t)\cos y\}\ dt holds. (1) Express f(x \plus{} y) in terms of

f(x),\ f(y).

(2) Find f'\left(x \plus{} \frac {\pi}{2}\right). Note that f\left(\frac {\pi}{2}\right) \equal{} 2. (3) Find

f(x).

1

Today's calculation of Integral 263

Let F(t)\equal{}\frac{1}{t}\int_0^{\frac{\pi}{2}t}|\cos 2x|\ dx for

0<t\leq 1.

\lim_{t\rightarrow 0} F(t).

(2) Find the range of

t

F(t)\geq 1.

1

Today's calculation of Integral 264

Find the area of the figure surrounded by the locus of the point

P

inside the square with side length

a

such that the distance to the center of the square is equal to the minimum distance to the side of the square.

1

Today's calculation of Integral 262

Answer the following questions for positive integer

n.

(1) Find the maximum value of f_n(x) \equal{} x^ne^{ \minus{} x} for

x\geq 0.

(2) Show that \lim_{x\to\infty} f_n(x) \equal{} 0. (3) Let I_n (x) \equal{} \int_0^x f_n(t)\ dt, find

\lim_{x\to\infty} I_n(x).

1

Today's calculation of Integral 261

Find the continuous function

f(x)

such that f(x)\equal{}\int_0^ x \{f(t)\cos t\minus{}\cos (t\minus{}x)\}\ dt.

1

Today's calculation of Integral 260

Evaluate \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(\sin ^3 \theta \minus{}\cos ^ 3 \theta \minus{}\cos ^ 2 \theta)(\sin \theta \plus{}\cos \theta \plus{}\cos ^ 2 \theta)^{2007}}{\sin ^ {2009}\theta \cos ^ {2009}\theta}\ d\theta

1

Today's calculation of Integral 259

Evaluate \int_0^{\frac{\pi}{12}} \frac{dx}{(\sin x\plus{}\cos x)^4}

1

Today's calculation of Integral 258

Find the volume of the solid formed by the revolution of the curve x \equal{} a(1 \plus{} \cos \theta )\cos \theta ,\ y \equal{} a(1 \plus{} \cos \theta)\sin \theta\ (a > 0,\ \minus{} \pi < \theta \leq \pi) about the

x

1

Today's calculation of Integral 257

\sqrt{\pi \int_0^{2008} x|\sin \pi x|\ dx}