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Today's Calculation Of Integral
2005 Today's Calculation Of Integral
2005 Today's Calculation Of Integral
Part of
Today's Calculation Of Integral
Subcontests
(91)
91
1
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Today's calculation of integral 91
Prove the following inequality.
∑
n
=
0
∞
∫
0
1
x
4011
(
1
−
x
2006
)
n
−
1
2006
d
x
<
2006
2005
\sum_{n=0}^\infty \int_0^1 x^{4011} (1-x^{2006})^\frac{n-1}{2006}\ dx<\frac{2006}{2005}
n
=
0
∑
∞
∫
0
1
x
4011
(
1
−
x
2006
)
2006
n
−
1
d
x
<
2005
2006
90
1
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Today's calculation of integral 90
Find
lim
n
→
∞
(
3
n
C
n
2
n
C
n
)
1
n
\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}
lim
n
→
∞
(
2
n
C
n
3
n
C
n
)
n
1
where
i
C
j
_iC_j
i
C
j
is a binominal coefficient which means
i
⋅
(
i
−
1
)
⋯
(
i
−
j
+
1
)
j
⋅
(
j
−
1
)
⋯
2
⋅
1
\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}
j
⋅
(
j
−
1
)
⋯
2
⋅
1
i
⋅
(
i
−
1
)
⋯
(
i
−
j
+
1
)
.
89
1
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Today's calculation of integral 89
For
f
(
x
)
=
x
4
+
∣
x
∣
,
f(x)=x^4+|x|,
f
(
x
)
=
x
4
+
∣
x
∣
,
let
I
1
=
∫
0
π
f
(
cos
x
)
d
x
,
I
2
=
∫
0
π
2
f
(
sin
x
)
d
x
.
I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.
I
1
=
∫
0
π
f
(
cos
x
)
d
x
,
I
2
=
∫
0
2
π
f
(
sin
x
)
d
x
.
Find the value of
I
1
I
2
.
\frac{I_1}{I_2}.
I
2
I
1
.
88
1
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Today's calculation of integral 88
A function
f
(
x
)
f(x)
f
(
x
)
satisfies
{
f
(
x
)
=
−
f
′
′
(
x
)
−
(
4
x
−
2
)
f
′
(
x
)
f
(
0
)
=
a
,
f
(
1
)
=
b
\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}
{
f
(
x
)
=
−
f
′′
(
x
)
−
(
4
x
−
2
)
f
′
(
x
)
f
(
0
)
=
a
,
f
(
1
)
=
b
Evaluate
∫
0
1
f
(
x
)
(
x
2
−
x
)
d
x
.
\int_0^1 f(x)(x^2-x)\ dx.
∫
0
1
f
(
x
)
(
x
2
−
x
)
d
x
.
87
1
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Today's calculation of integral 87
Find the minimum value of
a
(
0
<
a
<
1
)
a\ (0<a<1)
a
(
0
<
a
<
1
)
for which the following definite integral is minimized.
∫
0
π
∣
sin
x
−
a
x
∣
d
x
\int_0^{\pi} |\sin x-ax|\ dx
∫
0
π
∣
sin
x
−
a
x
∣
d
x
86
1
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Today's calculation of Integral 86
Prove
[
∫
π
∞
cos
x
x
d
x
]
2
<
1
π
2
\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}
[
∫
π
∞
x
cos
x
d
x
]
2
<
π
2
1
85
1
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Today's calculation of Integral 85
Evaluate
lim
n
→
∞
∫
0
π
2
[
n
sin
x
]
n
d
x
\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx
n
→
∞
lim
∫
0
2
π
n
[
n
sin
x
]
d
x
where
[
x
]
[x]
[
x
]
is the integer equal to
x
x
x
or less than
x
x
x
.
84
1
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Today's calculation of Integral 84
Evaluate
lim
n
→
∞
n
∫
0
π
e
−
n
x
sin
2
n
x
d
x
\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx
n
→
∞
lim
n
∫
0
π
e
−
n
x
sin
2
n
x
d
x
83
1
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Today's calculation of Integral 83
Evaluate
∑
n
=
1
∞
∫
2
n
π
2
(
n
+
1
)
π
x
sin
x
+
cos
x
x
2
d
x
(
n
=
1
,
2
,
⋯
)
\sum_{n=1}^{\infty} \int_{2n\pi}^{2(n+1)\pi} \frac{x\sin x+\cos x}{x^2}\ dx\ (n=1,2,\cdots)
n
=
1
∑
∞
∫
2
nπ
2
(
n
+
1
)
π
x
2
x
sin
x
+
cos
x
d
x
(
n
=
1
,
2
,
⋯
)
82
1
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Today's calculation of Integral 82
Let
0
<
a
<
b
0<a<b
0
<
a
<
b
.Prove the following inequaliy.
1
b
−
a
∫
a
b
(
ln
b
x
)
2
d
x
<
2
\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2
b
−
a
1
∫
a
b
(
ln
x
b
)
2
d
x
<
2
81
1
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Today's calculation of Integral 81
Prove the following inequality.
1
12
(
π
−
6
+
2
3
)
≤
∫
π
6
π
4
ln
(
1
+
cos
2
x
)
d
x
≤
1
4
(
2
−
3
)
\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})
12
1
(
π
−
6
+
2
3
)
≤
∫
6
π
4
π
ln
(
1
+
cos
2
x
)
d
x
≤
4
1
(
2
−
3
)
80
1
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Today's calculation of Integral 80
Let
S
S
S
be the domain surrounded by the two curves
C
1
:
y
=
a
x
2
,
C
2
:
y
=
−
a
x
2
+
2
a
b
x
C_1:y=ax^2,\ C_2:y=-ax^2+2abx
C
1
:
y
=
a
x
2
,
C
2
:
y
=
−
a
x
2
+
2
ab
x
for constant positive numbers
a
,
b
a,b
a
,
b
. Let
V
x
V_x
V
x
be the volume of the solid formed by the revolution of
S
S
S
about the axis of
x
x
x
,
V
y
V_y
V
y
be the volume of the solid formed by the revolution of
S
S
S
about the axis of
y
y
y
. Find the ratio of
V
x
V
y
\frac{V_x}{V_y}
V
y
V
x
.
79
1
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Today's calculation of Integral 79
Find the area of the domain expressed by the following system inequalities.
x
≥
0
,
y
≥
0
,
x
1
p
+
y
1
p
≤
1
(
p
=
1
,
2
,
⋯
)
x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)
x
≥
0
,
y
≥
0
,
x
p
1
+
y
p
1
≤
1
(
p
=
1
,
2
,
⋯
)
78
1
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Today's calculation of Integral 78
Let
α
,
β
\alpha,\beta
α
,
β
be the distinct positive roots of the equation of
2
x
=
tan
x
2x=\tan x
2
x
=
tan
x
. Evaluate
∫
0
1
sin
α
x
sin
β
x
d
x
\int_0^1 \sin \alpha x\sin \beta x\ dx
∫
0
1
sin
αx
sin
β
x
d
x
77
1
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Today's calculation of Integral 77
Find the area of the part enclosed by the following curve.
x
2
+
2
a
x
y
+
y
2
=
1
(
−
1
<
a
<
1
)
x^2+2axy+y^2=1\ (-1<a<1)
x
2
+
2
a
x
y
+
y
2
=
1
(
−
1
<
a
<
1
)
76
1
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Today's calculation of Integral 76
The function
f
n
(
x
)
(
n
=
1
,
2
,
⋯
)
f_n (x)\ (n=1,2,\cdots)
f
n
(
x
)
(
n
=
1
,
2
,
⋯
)
is defined as follows.
f
1
(
x
)
=
x
,
f
n
+
1
(
x
)
=
2
x
n
+
1
−
x
n
+
1
2
∫
0
1
f
n
(
t
)
d
t
(
n
=
1
,
2
,
⋯
)
f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)
f
1
(
x
)
=
x
,
f
n
+
1
(
x
)
=
2
x
n
+
1
−
x
n
+
2
1
∫
0
1
f
n
(
t
)
d
t
(
n
=
1
,
2
,
⋯
)
Evaluate
lim
n
→
∞
f
n
(
1
+
1
2
n
)
\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)
n
→
∞
lim
f
n
(
1
+
2
n
1
)
75
1
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Today's calculation of Integral 75
A function
f
(
θ
)
f(\theta)
f
(
θ
)
satisfies the following conditions
(
a
)
,
(
b
)
(a),(b)
(
a
)
,
(
b
)
.
(
a
)
f
(
θ
)
≥
0
(a)\ f(\theta)\geq 0
(
a
)
f
(
θ
)
≥
0
(
b
)
∫
0
π
f
(
θ
)
sin
θ
d
θ
=
1
(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1
(
b
)
∫
0
π
f
(
θ
)
sin
θ
d
θ
=
1
Prove the following inequality.
∫
0
π
f
(
θ
)
sin
n
θ
d
θ
≤
n
(
n
=
1
,
2
,
⋯
)
\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)
∫
0
π
f
(
θ
)
sin
n
θ
d
θ
≤
n
(
n
=
1
,
2
,
⋯
)
74
1
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Today's Calculation of Integral 74
p
,
q
p,q
p
,
q
satisfies
p
x
+
q
≥
ln
x
px+q\geq \ln x
p
x
+
q
≥
ln
x
at
a
≤
x
≤
b
(
0
<
a
<
b
)
a\leq x\leq b\ (0<a<b)
a
≤
x
≤
b
(
0
<
a
<
b
)
. Find the value of
p
,
q
p,q
p
,
q
for which the following definite integral is minimized and then the minimum value.
∫
a
b
(
p
x
+
q
−
ln
x
)
d
x
\int_a^b (px+q-\ln x)dx
∫
a
b
(
p
x
+
q
−
ln
x
)
d
x
73
1
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Today's Calculation of Integral 73
Find the minimum value of
∫
0
π
(
a
sin
x
+
b
sin
2
x
+
c
sin
3
x
−
x
)
2
d
x
\int_0^{\pi} (a\sin x+b\sin 2x+c\sin 3x-x)^2\ dx
∫
0
π
(
a
sin
x
+
b
sin
2
x
+
c
sin
3
x
−
x
)
2
d
x
72
1
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Today's Calculation of Integral 72
Let
f
(
x
)
f(x)
f
(
x
)
be a continuous function satisfying
f
(
x
)
=
1
+
k
∫
−
π
2
π
2
f
(
t
)
sin
(
x
−
t
)
d
t
(
k
:
c
o
n
s
t
a
n
t
n
u
m
b
e
r
)
f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)
f
(
x
)
=
1
+
k
∫
−
2
π
2
π
f
(
t
)
sin
(
x
−
t
)
d
t
(
k
:
co
n
s
t
an
t
n
u
mb
er
)
Find the value of
k
k
k
for which
∫
0
π
f
(
x
)
d
x
\int_0^{\pi} f(x)dx
∫
0
π
f
(
x
)
d
x
is maximized.
71
1
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Today's Calculation of Integral 71
Find the minimum value of
∫
−
1
1
∣
t
−
x
∣
d
t
\int_{-1}^1 \sqrt{|t-x|}\ dt
∫
−
1
1
∣
t
−
x
∣
d
t
70
1
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Today's Calculation of Integral 70
Find the number of root for
∫
0
π
2
e
x
cos
(
x
+
a
)
d
x
=
0
\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0
∫
0
2
π
e
x
cos
(
x
+
a
)
d
x
=
0
at
0
≤
a
<
2
π
0\leq a <2\pi
0
≤
a
<
2
π
69
1
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Today's Calculation of Integral 69
Let
f
1
(
x
)
=
x
,
f
n
(
x
)
=
x
+
1
14
∫
0
π
x
f
n
−
1
(
t
)
cos
3
t
d
t
(
n
≥
2
)
f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)
f
1
(
x
)
=
x
,
f
n
(
x
)
=
x
+
14
1
∫
0
π
x
f
n
−
1
(
t
)
cos
3
t
d
t
(
n
≥
2
)
. Find
lim
n
→
∞
f
n
(
x
)
\lim_{n\to\infty} f_n(x)
lim
n
→
∞
f
n
(
x
)
68
1
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Today's Calculation of Integral 68
Find the minimum value of
∫
1
e
∣
ln
x
−
a
x
∣
d
x
(
0
≤
a
≤
e
)
\int_1^e \left|\ln x-\frac{a}{x}\right|dx\ (0\leq a\leq e)
∫
1
e
ln
x
−
x
a
d
x
(
0
≤
a
≤
e
)
67
1
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Today's calculation of Integral 67
Evaluate
2005
∫
0
1002
d
x
100
2
2
−
x
2
+
100
3
2
−
x
2
+
∫
1002
1003
100
3
2
−
x
2
d
x
∫
0
1
1
−
x
2
d
x
\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}
∫
0
1
1
−
x
2
d
x
2005
∫
0
1002
100
2
2
−
x
2
+
100
3
2
−
x
2
d
x
+
∫
1002
1003
100
3
2
−
x
2
d
x
66
1
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Today's calculation of Integral 66
Find the minimum value of
∫
0
π
2
∣
cos
x
−
a
∣
sin
x
d
x
\int_0^{\frac{\pi}{2}} |\cos x -a|\sin x \ dx
∫
0
2
π
∣
cos
x
−
a
∣
sin
x
d
x
65
1
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Today's calculation of Integral 65
Let
a
>
0
a>0
a
>
0
. Find the minimum value of
∫
−
1
a
(
1
−
x
a
)
1
+
x
d
x
\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx
∫
−
1
a
(
1
−
a
x
)
1
+
x
d
x
64
1
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Today's calculation of Integral 64
Let
f
(
t
)
f(t)
f
(
t
)
be the cubic polynomial for
t
t
t
such that
cos
3
x
=
f
(
cos
x
)
\cos 3x=f(\cos x)
cos
3
x
=
f
(
cos
x
)
holds for all real number
x
x
x
. Evaluate
∫
0
1
{
f
(
t
)
}
2
1
−
t
2
d
t
\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt
∫
0
1
{
f
(
t
)
}
2
1
−
t
2
d
t
63
1
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Today's calculation of Integral 63
For a positive number
x
x
x
, let
f
(
x
)
=
lim
n
→
∞
∑
k
=
1
n
∣
cos
(
2
k
+
1
2
n
x
)
−
cos
(
2
k
−
1
2
n
x
)
∣
f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|
f
(
x
)
=
lim
n
→
∞
∑
k
=
1
n
cos
(
2
n
2
k
+
1
x
)
−
cos
(
2
n
2
k
−
1
x
)
Evaluate
lim
x
→
∞
f
(
x
)
x
\lim_{x\rightarrow\infty}\frac{f(x)}{x}
x
→
∞
lim
x
f
(
x
)
62
1
Hide problems
Today's Calculation of Integral 62
For
a
>
1
a>1
a
>
1
, let
f
(
a
)
=
1
2
∫
0
1
∣
a
x
n
−
1
∣
d
x
+
1
2
(
n
=
1
,
2
,
⋯
)
f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)
f
(
a
)
=
2
1
∫
0
1
∣
a
x
n
−
1∣
d
x
+
2
1
(
n
=
1
,
2
,
⋯
)
and let
b
n
b_n
b
n
be the minimum value of
f
(
a
)
f(a)
f
(
a
)
at
a
>
1
a>1
a
>
1
. Evaluate
lim
m
→
∞
b
m
⋅
b
m
+
1
⋅
⋯
⋯
b
2
m
(
m
=
1
,
2
,
3
,
⋯
)
\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)
m
→
∞
lim
b
m
⋅
b
m
+
1
⋅
⋯⋯
b
2
m
(
m
=
1
,
2
,
3
,
⋯
)
61
1
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Today's Calculation of Integral 61
Evaluate
∑
k
=
0
2004
∫
−
1
1
1
−
x
2
k
+
1
−
x
d
x
\sum_{k=0}^{2004} \int_{-1}^1 \frac{\sqrt{1-x^2}}{\sqrt{k+1}-x}dx
k
=
0
∑
2004
∫
−
1
1
k
+
1
−
x
1
−
x
2
d
x
60
1
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Today's Calculation of Integral 60
Let
a
n
=
∫
0
π
2
sin
2
t
(
1
−
sin
t
)
n
−
1
2
d
t
(
n
=
1
,
2
,
⋯
)
a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)
a
n
=
∫
0
2
π
sin
2
t
(
1
−
sin
t
)
2
n
−
1
d
t
(
n
=
1
,
2
,
⋯
)
Evaluate
∑
n
=
1
∞
(
n
+
1
)
(
a
n
−
a
n
+
1
)
\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})
n
=
1
∑
∞
(
n
+
1
)
(
a
n
−
a
n
+
1
)
59
1
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Today's Calculation of Integral 59
Evaluate
∫
−
π
π
(
cos
2
x
)
(
cos
2
2
x
)
⋯
(
cos
2
2006
x
)
d
x
\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx
∫
−
π
π
(
cos
2
x
)
(
cos
2
2
x
)
⋯
(
cos
2
2006
x
)
d
x
58
1
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Today's Calculation of Integral 58
Let
f
(
x
)
=
e
x
e
x
+
1
f(x)=\frac{e^x}{e^x+1}
f
(
x
)
=
e
x
+
1
e
x
Prove the following equation.
∫
a
b
f
(
x
)
d
x
+
∫
f
(
a
)
f
(
b
)
f
−
1
(
x
)
d
x
=
b
f
(
b
)
−
a
f
(
a
)
\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)
∫
a
b
f
(
x
)
d
x
+
∫
f
(
a
)
f
(
b
)
f
−
1
(
x
)
d
x
=
b
f
(
b
)
−
a
f
(
a
)
57
1
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Today's Calculation of Integral 57
Find the value of
n
∈
N
n\in{\mathbb{N}}
n
∈
N
satisfying the following inequality.
∣
∫
0
π
x
2
sin
n
x
d
x
∣
<
99
π
2
100
n
\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}
∫
0
π
x
2
sin
n
x
d
x
<
100
n
99
π
2
56
1
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Today's Calculation of Integaral 56
Evaluate
lim
n
→
∞
∑
k
=
1
n
[
2
n
2
−
k
2
]
n
2
\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}
n
→
∞
lim
k
=
1
∑
n
n
2
[
2
n
2
−
k
2
]
[
x
]
[x]
[
x
]
is the greatest integer
≤
x
\leq x
≤
x
.
55
1
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Today's Calculation of Integral 55
Evaluate
lim
n
→
∞
n
∫
0
1
(
1
+
x
)
−
n
−
1
e
x
2
d
x
(
n
=
1
,
2
,
⋯
)
\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)
n
→
∞
lim
n
∫
0
1
(
1
+
x
)
−
n
−
1
e
x
2
d
x
(
n
=
1
,
2
,
⋯
)
54
1
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Today's Calculation of Integral 54
evaluate
∫
−
1
0
1
+
x
1
−
x
d
x
\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx
∫
−
1
0
1
−
x
1
+
x
d
x
53
1
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Today's Calculation of Integral 53
Find the maximum value of the following integral.
∫
0
∞
e
−
x
sin
t
x
d
x
\int_0^{\infty} e^{-x}\sin tx\ dx
∫
0
∞
e
−
x
sin
t
x
d
x
52
1
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Today's Calculation of Integarl 52
Evaluate
lim
n
→
∞
∑
k
=
1
n
1
n
+
k
−
1
\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}
n
→
∞
lim
k
=
1
∑
n
n
+
k
−
1
1
51
1
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Today's Calculation of Integral 51
A function
f
(
x
)
f(x)
f
(
x
)
satisfies
f
(
x
)
=
f
(
c
x
)
f(x)=f\left(\frac{c}{x}\right)
f
(
x
)
=
f
(
x
c
)
for some real number
c
(
>
1
)
c(>1)
c
(
>
1
)
and all positive number
x
x
x
. If
∫
1
c
f
(
x
)
x
d
x
=
3
\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3
∫
1
c
x
f
(
x
)
d
x
=
3
, evaluate
∫
1
c
f
(
x
)
x
d
x
\int_1^c \frac{f(x)}{x} dx
∫
1
c
x
f
(
x
)
d
x
50
1
Hide problems
Today's Calculation of Integral 50
Let
a
,
b
a,b
a
,
b
be real numbers such that
a
<
b
a<b
a
<
b
. Evaluate
lim
b
→
a
∫
a
b
ln
∣
1
+
(
x
−
a
)
(
b
−
x
)
∣
d
x
(
b
−
a
)
3
\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}
b
→
a
lim
(
b
−
a
)
3
∫
a
b
ln
∣1
+
(
x
−
a
)
(
b
−
x
)
∣
d
x
.
49
1
Hide problems
Today's Calculation of Integral 49
For
x
≥
0
x\geq 0
x
≥
0
, Prove that
∫
0
x
(
t
−
t
2
)
sin
2002
t
d
t
<
1
2004
⋅
2005
\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}
∫
0
x
(
t
−
t
2
)
sin
2002
t
d
t
<
2004
⋅
2005
1
48
1
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Today's Calculation of Integral 48
Evaluate
lim
n
→
∞
(
∫
0
π
sin
2
n
x
sin
x
d
x
−
∑
k
=
1
n
1
k
)
\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)
n
→
∞
lim
(
∫
0
π
sin
x
sin
2
n
x
d
x
−
k
=
1
∑
n
k
1
)
47
1
Hide problems
Today's Calculation of Integral 47
Find the condition of
a
,
b
a,b
a
,
b
for which the function
f
(
x
)
(
0
≤
x
≤
2
π
)
f(x)\ (0\leq x\leq 2\pi)
f
(
x
)
(
0
≤
x
≤
2
π
)
satisfying the following equality can be determined uniquely,then determine
f
(
x
)
f(x)
f
(
x
)
, assuming that
f
(
x
)
f(x)
f
(
x
)
is a continuous function at
0
≤
x
≤
2
π
0\leq x\leq 2\pi
0
≤
x
≤
2
π
.
f
(
x
)
=
a
2
π
∫
0
2
π
sin
(
x
+
y
)
f
(
y
)
d
y
+
b
2
π
∫
0
2
π
cos
(
x
−
y
)
f
(
y
)
d
y
+
sin
x
+
cos
x
f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x
f
(
x
)
=
2
π
a
∫
0
2
π
sin
(
x
+
y
)
f
(
y
)
d
y
+
2
π
b
∫
0
2
π
cos
(
x
−
y
)
f
(
y
)
d
y
+
sin
x
+
cos
x
46
1
Hide problems
Today's calculation of Integral 46
Find the minimum value of
∫
0
1
∣
t
−
x
∣
t
+
1
d
t
\int_0^1 \frac{|t-x|}{t+1}dt
∫
0
1
t
+
1
∣
t
−
x
∣
d
t
45
1
Hide problems
Today's calculation of Integral 45
Find the function
f
(
x
)
f(x)
f
(
x
)
which satisfies the following integral equation.
f
(
x
)
=
∫
0
x
t
(
sin
t
−
cos
t
)
d
t
+
∫
0
π
2
e
t
f
(
t
)
d
t
f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt
f
(
x
)
=
∫
0
x
t
(
sin
t
−
cos
t
)
d
t
+
∫
0
2
π
e
t
f
(
t
)
d
t
44
1
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Today's calculation of Integral 44
Evaluate
∫
0
π
2
sin
2005
x
sin
x
d
x
{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx
∫
0
2
π
sin
x
sin
2005
x
d
x
43
1
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Today's calculation of Integral 43
Evaluate
∫
0
π
2
cos
2004
x
cos
2004
x
d
x
\int_0^{\frac{\pi}{2}} \cos ^ {2004}x\cos 2004x\ dx
∫
0
2
π
cos
2004
x
cos
2004
x
d
x
42
1
Hide problems
Today's calculation of Integral 42
Let
0
<
t
<
π
2
0<t<\frac{\pi}{2}
0
<
t
<
2
π
. Evaluate
lim
t
→
π
2
∫
0
t
tan
θ
cos
θ
ln
(
cos
θ
)
d
θ
\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta
t
→
2
π
lim
∫
0
t
tan
θ
cos
θ
ln
(
cos
θ
)
d
θ
41
1
Hide problems
Today's calculation of Integral 41
Evaluate
∫
0
a
2
a
x
−
x
2
d
x
(
a
>
0
)
\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)
∫
0
a
2
a
x
−
x
2
d
x
(
a
>
0
)
40
1
Hide problems
Today's calculation of Integral 40
Evaluate
∫
0
1
x
2005
e
−
x
2
d
x
\int_0^1 x^{2005}e^{-x^2}dx
∫
0
1
x
2005
e
−
x
2
d
x
39
1
Hide problems
Today's calculation of Integral 39
Find the minimum value of the following function
f
(
x
)
f(x)
f
(
x
)
defined at
0
<
x
<
π
2
0<x<\frac{\pi}{2}
0
<
x
<
2
π
.
f
(
x
)
=
∫
0
x
d
θ
cos
θ
+
∫
x
π
2
d
θ
sin
θ
f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}
f
(
x
)
=
∫
0
x
cos
θ
d
θ
+
∫
x
2
π
sin
θ
d
θ
38
1
Hide problems
Today's calculation of Integral 38
Let
a
a
a
be a constant number such that
0
<
a
<
1
0<a<1
0
<
a
<
1
and
V
(
a
)
V(a)
V
(
a
)
be the volume formed by the revolution of the figure which is enclosed by the curve
y
=
ln
(
x
−
a
)
y=\ln (x-a)
y
=
ln
(
x
−
a
)
, the
x
x
x
-axis and two lines
x
=
1
,
x
=
3
x=1,x=3
x
=
1
,
x
=
3
about the
x
x
x
-axis. If
a
a
a
varies in the range of
0
<
a
<
1
0<a<1
0
<
a
<
1
, find the minimum value of
V
(
a
)
V(a)
V
(
a
)
.
37
1
Hide problems
Today's calculation of Integral 37
Evaluate
∫
π
2
2
π
3
1
sin
x
1
−
cos
x
d
x
\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx
∫
2
π
3
2
π
sin
x
1
−
cos
x
1
d
x
36
1
Hide problems
Today's calculation of Integral 36
A sequence of polynomial
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
f_n(x)\ (n=0,1,2,\cdots)
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
satisfies
f
0
(
x
)
=
2
,
f
1
(
x
)
=
x
f_0(x)=2,f_1(x)=x
f
0
(
x
)
=
2
,
f
1
(
x
)
=
x
,
f
n
(
x
)
=
x
f
n
−
1
(
x
)
−
f
n
−
2
(
x
)
,
(
n
=
2
,
3
,
4
,
⋯
)
f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)
f
n
(
x
)
=
x
f
n
−
1
(
x
)
−
f
n
−
2
(
x
)
,
(
n
=
2
,
3
,
4
,
⋯
)
Let
x
n
(
n
≧
2
)
x_n\ (n\geqq 2)
x
n
(
n
≧
2
)
be the maximum real root of the equation
f
n
(
x
)
=
0
(
∣
x
∣
≦
2
)
f_n(x)=0\ (|x|\leqq 2)
f
n
(
x
)
=
0
(
∣
x
∣
≦
2
)
Evaluate
lim
n
→
∞
n
2
∫
x
n
2
f
n
(
x
)
d
x
\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx
n
→
∞
lim
n
2
∫
x
n
2
f
n
(
x
)
d
x
35
1
Hide problems
Today's calculation of Integral 35
Determine the value of
a
,
b
a,b
a
,
b
for which
∫
0
1
(
1
−
x
−
a
x
−
b
)
2
d
x
\int_0^1 (\sqrt{1-x}-ax-b)^2 dx
∫
0
1
(
1
−
x
−
a
x
−
b
)
2
d
x
is minimized.
34
1
Hide problems
Today's calculation of Integral 34
Let
p
p
p
be a constant number such that
0
<
p
<
1
0<p<1
0
<
p
<
1
. Evaluate
∑
k
=
0
2004
p
k
(
1
−
p
)
2004
−
k
∫
0
1
x
k
(
1
−
x
)
2004
−
k
d
x
\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}
k
=
0
∑
2004
∫
0
1
x
k
(
1
−
x
)
2004
−
k
d
x
p
k
(
1
−
p
)
2004
−
k
33
1
Hide problems
Today's calculation of Integral 33
Evaluate
∫
−
ln
2
0
d
x
cos
2
h
x
⋅
1
−
2
a
tanh
x
+
a
2
(
a
>
0
)
\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)
∫
−
l
n
2
0
cos
2
h
x
⋅
1
−
2
a
tanh
x
+
a
2
d
x
(
a
>
0
)
32
1
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Today's calculation of Integral 32
Evaluate
∫
0
1
e
x
+
e
x
+
e
e
x
+
e
e
e
x
d
x
\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx
∫
0
1
e
x
+
e
x
+
e
e
x
+
e
e
e
x
d
x
31
1
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Today's calculation of Integral 31
Evaluate
lim
n
→
∞
∫
0
π
x
2
∣
sin
n
x
∣
d
x
\lim_{n\to\infty} \int_0^{\pi} x^2 |\sin nx| dx
n
→
∞
lim
∫
0
π
x
2
∣
sin
n
x
∣
d
x
30
1
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Today's calculation of Integral 30
A sequence
{
a
n
}
\{a_n\}
{
a
n
}
is defined by
a
n
=
∫
0
1
x
3
(
1
−
x
)
n
d
x
(
n
=
1
,
2
,
3.
⋯
)
a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)
a
n
=
∫
0
1
x
3
(
1
−
x
)
n
d
x
(
n
=
1
,
2
,
3.
⋯
)
Find the constant number
c
c
c
such that
∑
n
=
1
∞
(
n
+
c
)
(
a
n
−
a
n
+
1
)
=
1
3
\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}
∑
n
=
1
∞
(
n
+
c
)
(
a
n
−
a
n
+
1
)
=
3
1
29
1
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Today's calculation of Integral 29
Let
a
a
a
be a real number. Evaluate
∫
−
π
+
a
3
π
+
a
∣
x
−
a
−
π
∣
sin
(
x
2
)
d
x
\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx
∫
−
π
+
a
3
π
+
a
∣
x
−
a
−
π
∣
sin
(
2
x
)
d
x
28
1
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Today's calculation of Integral 28
Evaluate
∫
0
π
4
x
cos
5
x
cos
x
d
x
\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx
∫
0
4
π
cos
x
x
cos
5
x
d
x
27
1
Hide problems
Today's calculation of Integral 27
Let
f
(
x
)
=
t
sin
x
+
(
1
−
t
)
cos
x
(
0
≦
t
≦
1
)
f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)
f
(
x
)
=
t
sin
x
+
(
1
−
t
)
cos
x
(
0
≦
t
≦
1
)
. Find the maximum and minimum value of the following
P
(
t
)
P(t)
P
(
t
)
.
P
(
t
)
=
{
∫
0
π
2
e
x
f
(
x
)
d
x
}
{
∫
0
π
2
e
−
x
f
(
x
)
d
x
}
P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}
P
(
t
)
=
{
∫
0
2
π
e
x
f
(
x
)
d
x
}
{
∫
0
2
π
e
−
x
f
(
x
)
d
x
}
26
1
Hide problems
Today's calculation of Integral 26
Evaluate
∫
e
e
e
e
e
e
e
d
x
x
ln
x
⋅
ln
(
ln
x
)
⋅
ln
{
ln
(
ln
x
)
}
{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}
∫
e
e
e
e
e
e
e
x
ln
x
⋅
ln
(
ln
x
)
⋅
ln
{
ln
(
ln
x
)}
d
x
25
1
Hide problems
Today's calculation of Integral 25
Let
∣
a
∣
<
π
2
|a|<\frac{\pi}{2}
∣
a
∣
<
2
π
. Evaluate
∫
0
π
2
d
x
{
sin
(
a
+
x
)
+
cos
x
}
2
\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}
∫
0
2
π
{
sin
(
a
+
x
)
+
cos
x
}
2
d
x
24
1
Hide problems
Today's calculation of Integral 24
Find the minimum value of
∫
0
π
(
x
−
y
)
2
(
sin
x
)
∣
cos
x
∣
d
x
\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx
∫
0
π
(
x
−
y
)
2
(
sin
x
)
∣
cos
x
∣
d
x
.
23
1
Hide problems
Today's calculation of Integral 23
Evaluate
lim
a
→
π
2
−
0
∫
0
a
(
cos
x
)
ln
(
cos
x
)
d
x
(
0
≦
a
<
π
2
)
\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)
a
→
2
π
−
0
lim
∫
0
a
(
cos
x
)
ln
(
cos
x
)
d
x
(
0
≦
a
<
2
π
)
22
1
Hide problems
Today's calculation of Integral 22
Evaluate
∫
0
1
(
1
−
x
2
)
n
d
x
(
n
=
0
,
1
,
2
,
⋯
)
\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)
∫
0
1
(
1
−
x
2
)
n
d
x
(
n
=
0
,
1
,
2
,
⋯
)
21
1
Hide problems
Today's calculation of Integral 21
[1] Tokyo Univ. of Science:
∫
ln
x
(
x
+
1
)
2
d
x
\int \frac{\ln x}{(x+1)^2}dx
∫
(
x
+
1
)
2
l
n
x
d
x
[2] Saitama Univ.:
∫
5
3
sin
x
+
4
cos
x
d
x
\int \frac{5}{3\sin x+4\cos x}dx
∫
3
s
i
n
x
+
4
c
o
s
x
5
d
x
[3] Yokohama City Univ.:
∫
1
3
1
x
2
+
1
d
x
\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx
∫
1
3
x
2
+
1
1
d
x
[4] Daido Institute of Technology:
∫
0
π
2
sin
3
x
sin
x
+
cos
x
d
x
\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx
∫
0
2
π
s
i
n
x
+
c
o
s
x
s
i
n
3
x
d
x
[5] Gunma Univ.:
∫
0
3
π
4
{
(
1
+
x
)
sin
x
+
(
1
−
x
)
cos
x
}
d
x
\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx
∫
0
4
3
π
{(
1
+
x
)
sin
x
+
(
1
−
x
)
cos
x
}
d
x
20
1
Hide problems
Today's calculation of Integral 20
Calculate the following indefinite integrals. [1]
∫
ln
(
x
2
−
1
)
d
x
\int \ln (x^2-1)dx
∫
ln
(
x
2
−
1
)
d
x
[2]
∫
1
e
x
+
1
d
x
\int \frac{1}{e^x+1}dx
∫
e
x
+
1
1
d
x
[3]
∫
(
a
x
2
+
b
x
+
c
)
e
m
x
d
x
(
a
b
c
m
≠
0
)
\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)
∫
(
a
x
2
+
b
x
+
c
)
e
m
x
d
x
(
ab
c
m
=
0
)
[4]
∫
(
tan
x
+
1
tan
x
)
2
d
x
\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx
∫
(
tan
x
+
t
a
n
x
1
)
2
d
x
[5]
∫
1
−
sin
x
d
x
\int \sqrt{1-\sin x}dx
∫
1
−
sin
x
d
x
19
1
Hide problems
Today's calculation of Integral 19
Calculate the following indefinite integrals. [1]
∫
tan
3
x
d
x
\int \tan ^ 3 x dx
∫
tan
3
x
d
x
[2]
∫
a
m
x
+
n
d
x
(
a
>
0
,
a
≠
1
,
m
n
≠
0
)
\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)
∫
a
m
x
+
n
d
x
(
a
>
0
,
a
=
1
,
mn
=
0
)
[3]
∫
cos
5
x
d
x
\int \cos ^ 5 x dx
∫
cos
5
x
d
x
[4]
∫
sin
2
x
cos
3
x
d
x
\int \sin ^ 2 x\cos ^ 3 x dx
∫
sin
2
x
cos
3
x
d
x
[5]
∫
d
x
sin
x
\int \frac{dx}{\sin x}
∫
s
i
n
x
d
x
18
1
Hide problems
Today's calculation of Integral 18
Calculate the following indefinite integrals. [1]
∫
(
sin
x
+
cos
x
)
4
d
x
\int (\sin x+\cos x)^4 dx
∫
(
sin
x
+
cos
x
)
4
d
x
[2]
∫
e
2
x
e
x
+
1
d
x
\int \frac{e^{2x}}{e^x+1}dx
∫
e
x
+
1
e
2
x
d
x
[3]
∫
sin
4
x
d
x
\int \sin ^ 4 xdx
∫
sin
4
x
d
x
[4]
∫
sin
6
x
cos
2
x
d
x
\int \sin 6x\cos 2xdx
∫
sin
6
x
cos
2
x
d
x
[5]
∫
x
2
(
x
+
1
)
3
d
x
\int \frac{x^2}{\sqrt{(x+1)^3}}dx
∫
(
x
+
1
)
3
x
2
d
x
17
1
Hide problems
Today's calculation of Integral 17
Calculate the following indefinite integrals. [1]
∫
d
x
e
x
−
e
−
x
\int \frac{dx}{e^x-e^{-x}}
∫
e
x
−
e
−
x
d
x
[2]
∫
e
−
a
x
cos
2
x
d
x
(
a
≠
0
)
\int e^{-ax}\cos 2x dx\ (a\neq 0)
∫
e
−
a
x
cos
2
x
d
x
(
a
=
0
)
[3]
∫
(
3
x
−
2
)
2
d
x
\int (3^x-2)^2 dx
∫
(
3
x
−
2
)
2
d
x
[4]
∫
x
4
+
2
x
3
+
3
x
2
+
1
(
x
+
2
)
5
d
x
\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx
∫
(
x
+
2
)
5
x
4
+
2
x
3
+
3
x
2
+
1
d
x
[5]
∫
d
x
1
−
cos
x
d
x
\int \frac{dx}{1-\cos x}dx
∫
1
−
c
o
s
x
d
x
d
x
16
1
Hide problems
Today's calculation of Integral 16
Calculate the following indefinite integrals. [1]
∫
sin
(
ln
x
)
d
x
\int \sin (\ln x)dx
∫
sin
(
ln
x
)
d
x
[2]
∫
x
+
sin
2
x
x
sin
2
x
d
x
\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx
∫
x
s
i
n
2
x
x
+
s
i
n
2
x
d
x
[3]
∫
x
3
x
2
+
1
d
x
\int \frac{x^3}{x^2+1}dx
∫
x
2
+
1
x
3
d
x
[4]
∫
x
2
2
x
−
1
d
x
\int \frac{x^2}{\sqrt{2x-1}}dx
∫
2
x
−
1
x
2
d
x
[5]
∫
x
+
cos
2
x
+
1
x
cos
2
x
d
x
\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx
∫
x
c
o
s
2
x
x
+
c
o
s
2
x
+
1
d
x
15
1
Hide problems
Today's calculation of Integral 15
Calculate the following indefinite integrals. [1]
∫
(
x
2
−
1
)
2
x
4
d
x
\int \frac{(x^2-1)^2}{x^4}dx
∫
x
4
(
x
2
−
1
)
2
d
x
[2]
∫
e
3
x
e
x
+
1
d
x
\int \frac{e^{3x}}{\sqrt{e^x+1}}dx
∫
e
x
+
1
e
3
x
d
x
[3]
∫
sin
2
x
cos
3
x
d
x
\int \sin 2x\cos 3xdx
∫
sin
2
x
cos
3
x
d
x
[4]
∫
x
ln
(
x
+
1
)
d
x
\int x\ln (x+1)dx
∫
x
ln
(
x
+
1
)
d
x
[5]
∫
x
(
x
+
3
)
2
d
x
\int \frac{x}{(x+3)^2}dx
∫
(
x
+
3
)
2
x
d
x
14
1
Hide problems
Today's calculation of Integral 14
Calculate the following indefinite integrals. [1]
∫
sin
x
cos
x
1
+
sin
2
x
d
x
\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx
∫
1
+
s
i
n
2
x
s
i
n
x
c
o
s
x
d
x
[2]
∫
x
log
10
x
d
x
\int x\log_{10} x dx
∫
x
lo
g
10
x
d
x
[3]
∫
x
2
x
−
1
d
x
\int \frac{x}{\sqrt{2x-1}}dx
∫
2
x
−
1
x
d
x
[4]
∫
(
x
2
+
1
)
ln
x
d
x
\int (x^2+1)\ln x dx
∫
(
x
2
+
1
)
ln
x
d
x
[5]
∫
e
x
cos
x
d
x
\int e^x\cos x dx
∫
e
x
cos
x
d
x
13
1
Hide problems
Today's calculation of Integral 13
Calculate the following integarls. [1]
∫
x
cos
2
x
d
x
\int x\cos ^ 2 x dx
∫
x
cos
2
x
d
x
[2]
∫
x
−
1
(
3
x
−
1
)
2
d
x
\int \frac{x-1}{(3x-1)^2}dx
∫
(
3
x
−
1
)
2
x
−
1
d
x
[3]
∫
x
3
(
2
−
x
2
)
4
d
x
\int \frac{x^3}{(2-x^2)^4}dx
∫
(
2
−
x
2
)
4
x
3
d
x
[4]
∫
(
1
4
x
+
1
2
x
)
d
x
\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx
∫
(
4
x
1
+
2
x
1
)
d
x
[5]
∫
(
ln
x
)
2
d
x
\int (\ln x)^2 dx
∫
(
ln
x
)
2
d
x
12
1
Hide problems
Today's calculation of Integral 12
Calculate the following indefinite integrals. [1]
∫
d
x
1
+
cos
x
\int \frac{dx}{1+\cos x}
∫
1
+
c
o
s
x
d
x
[2]
∫
x
x
2
−
1
d
x
\int x\sqrt{x^2-1}dx
∫
x
x
2
−
1
d
x
[3]
∫
a
−
x
2
d
x
(
a
>
0
,
a
≠
1
)
\int a^{-\frac{x}{2}}dx\ \ (a>0,a\neq 1)
∫
a
−
2
x
d
x
(
a
>
0
,
a
=
1
)
[4]
∫
sin
3
x
1
+
cos
x
d
x
\int \frac{\sin ^ 3 x}{1+\cos x}dx
∫
1
+
c
o
s
x
s
i
n
3
x
d
x
[5]
∫
e
4
x
sin
2
x
d
x
\int e^{4x}\sin 2x dx
∫
e
4
x
sin
2
x
d
x
11
1
Hide problems
Today's calculation of Integral 11
Calculate the following indefinite integrals. [1]
∫
6
x
+
1
3
x
2
+
x
+
4
d
x
\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx
∫
3
x
2
+
x
+
4
6
x
+
1
d
x
[2]
∫
e
x
e
x
+
e
a
−
x
d
x
\int \frac{e^x}{e^x+e^{a-x}}dx
∫
e
x
+
e
a
−
x
e
x
d
x
[3]
∫
(
x
+
1
)
3
x
d
x
\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx
∫
x
(
x
+
1
)
3
d
x
[4]
∫
x
ln
(
x
2
−
1
)
d
x
\int x\ln (x^2-1)dx
∫
x
ln
(
x
2
−
1
)
d
x
[5]
∫
2
(
x
+
2
)
x
2
+
4
x
+
1
d
x
\int \frac{2(x+2)}{x^2+4x+1}dx
∫
x
2
+
4
x
+
1
2
(
x
+
2
)
d
x
10
1
Hide problems
Today's calculation of Integral 10
Calculate the following indefinite integrals. [1]
∫
(
2
x
+
1
)
x
+
2
d
x
\int (2x+1)\sqrt{x+2}\ dx
∫
(
2
x
+
1
)
x
+
2
d
x
[2]
∫
1
+
cos
x
x
+
sin
x
d
x
\int \frac{1+\cos x}{x+\sin x}\ dx
∫
x
+
s
i
n
x
1
+
c
o
s
x
d
x
[3]
∫
sin
5
x
cos
3
x
d
x
\int \sin ^ 5 x \cos ^ 3 x \ dx
∫
sin
5
x
cos
3
x
d
x
[4]
∫
(
x
−
3
)
2
x
4
d
x
\int \frac{(x-3)^2}{x^4}\ dx
∫
x
4
(
x
−
3
)
2
d
x
[5]
∫
d
x
tan
x
d
x
\int \frac{dx}{\tan x}\ dx
∫
t
a
n
x
d
x
d
x
9
1
Hide problems
Today's calculation of Integral 9
Calculate the following indefinite integrals. [1]
∫
(
x
2
+
4
x
−
3
)
2
(
x
+
2
)
d
x
\int (x^2+4x-3)^2(x+2)dx
∫
(
x
2
+
4
x
−
3
)
2
(
x
+
2
)
d
x
[2]
∫
ln
x
x
(
ln
x
+
1
)
d
x
\int \frac{\ln x}{x(\ln x+1)}dx
∫
x
(
l
n
x
+
1
)
l
n
x
d
x
[3]
∫
sin
(
π
log
2
x
)
x
d
x
\int \frac{\sin \ (\pi \log _2 x)}{x}dx
∫
x
s
i
n
(
π
l
o
g
2
x
)
d
x
[4]
∫
d
x
sin
x
cos
2
x
\int \frac{dx}{\sin x\cos ^ 2 x}
∫
s
i
n
x
c
o
s
2
x
d
x
[5]
∫
1
−
3
x
d
x
\int \sqrt{1-3x}\ dx
∫
1
−
3
x
d
x
8
1
Hide problems
Today's calculation of Integral 8
Calculate the following indefinite integrals. [1]
∫
x
(
x
2
+
3
)
2
d
x
\int x(x^2+3)^2 dx
∫
x
(
x
2
+
3
)
2
d
x
[2]
∫
ln
(
x
+
2
)
d
x
\int \ln (x+2) dx
∫
ln
(
x
+
2
)
d
x
[3]
∫
x
cos
x
d
x
\int x\cos x dx
∫
x
cos
x
d
x
[4]
∫
d
x
(
x
+
2
)
2
d
x
\int \frac{dx}{(x+2)^2}dx
∫
(
x
+
2
)
2
d
x
d
x
[5]
∫
x
−
1
x
2
−
2
x
+
3
d
x
\int \frac{x-1}{x^2-2x+3}dx
∫
x
2
−
2
x
+
3
x
−
1
d
x
7
1
Hide problems
Today's calculation of Integral 7
Calculate the following indefinite integrals. [1]
∫
x
(
x
+
1
)
2
d
x
\int \sqrt{x}(\sqrt{x}+1)^2 dx
∫
x
(
x
+
1
)
2
d
x
[2]
∫
(
e
x
+
2
e
x
+
1
−
3
e
x
+
2
)
d
x
\int (e^x+2e^{x+1}-3e^{x+2})dx
∫
(
e
x
+
2
e
x
+
1
−
3
e
x
+
2
)
d
x
[3]
∫
(
sin
2
x
+
cos
x
)
sin
x
d
x
\int (\sin ^2 x+\cos x)\sin x dx
∫
(
sin
2
x
+
cos
x
)
sin
x
d
x
[4]
∫
x
2
−
x
d
x
\int x\sqrt{2-x} dx
∫
x
2
−
x
d
x
[5]
∫
x
ln
x
d
x
\int x\ln x dx
∫
x
ln
x
d
x
6
1
Hide problems
Today's calculation of Integral 6
Calculate the following indefinite integrals. [1]
∫
sin
x
cos
3
x
d
x
\int \sin x\cos ^ 3 x dx
∫
sin
x
cos
3
x
d
x
[2]
∫
d
x
(
1
+
x
)
x
d
x
\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx
∫
(
1
+
x
)
x
d
x
d
x
[3]
∫
x
2
x
3
+
1
d
x
\int x^2 \sqrt{x^3+1}dx
∫
x
2
x
3
+
1
d
x
[4]
∫
e
2
x
−
3
e
x
e
x
d
x
\int \frac{e^{2x}-3e^{x}}{e^x}dx
∫
e
x
e
2
x
−
3
e
x
d
x
[5]
∫
(
1
−
x
2
)
e
x
d
x
\int (1-x^2)e^x dx
∫
(
1
−
x
2
)
e
x
d
x
5
1
Hide problems
Today's calculation of Integral 5
Calculate the following indefinite integrals. [1]
∫
(
4
−
5
tan
x
)
cos
x
d
x
\int (4-5\tan x)\cos x dx
∫
(
4
−
5
tan
x
)
cos
x
d
x
[2]
∫
d
x
(
1
−
3
x
)
2
3
d
x
\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx
∫
3
(
1
−
3
x
)
2
d
x
d
x
[3]
∫
x
3
4
−
x
2
d
x
\int x^3\sqrt{4-x^2}dx
∫
x
3
4
−
x
2
d
x
[4]
∫
e
−
x
sin
(
x
+
π
4
)
d
x
\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx
∫
e
−
x
sin
(
x
+
4
π
)
d
x
[5]
∫
(
3
x
−
4
)
2
d
x
\int (3x-4)^2 dx
∫
(
3
x
−
4
)
2
d
x
4
1
Hide problems
Today's calculation of Integral 4
Calculate the following indefinite integrals. [1]
∫
x
5
−
x
d
x
\int \frac{x}{\sqrt{5-x}}dx
∫
5
−
x
x
d
x
[2]
∫
sin
x
cos
2
x
1
+
cos
x
d
x
\int \frac{\sin x \cos ^2 x}{1+\cos x}dx
∫
1
+
c
o
s
x
s
i
n
x
c
o
s
2
x
d
x
[3]
∫
(
sin
x
+
cos
x
)
2
d
x
\int (\sin x+\cos x)^2dx
∫
(
sin
x
+
cos
x
)
2
d
x
[4]
∫
x
−
cos
2
x
x
cos
2
x
d
x
\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx
∫
x
c
o
s
2
x
x
−
c
o
s
2
x
d
x
[5]
∫
(
sin
x
+
sin
2
x
)
2
d
x
\int (\sin x+\sin 2x)^2 dx
∫
(
sin
x
+
sin
2
x
)
2
d
x
3
1
Hide problems
Today's calculation of Integral 3
Calculate the following indefinite integrals. [1]
∫
sin
x
sin
2
x
d
x
\int \sin x\sin 2x dx
∫
sin
x
sin
2
x
d
x
[2]
∫
e
2
x
e
x
−
1
d
x
\int \frac{e^{2x}}{e^x-1}dx
∫
e
x
−
1
e
2
x
d
x
[3]
∫
tan
2
x
cos
2
x
d
x
\int \frac{\tan ^2 x}{\cos ^2 x}dx
∫
c
o
s
2
x
t
a
n
2
x
d
x
[4]
∫
e
x
+
e
−
x
e
x
−
e
−
x
d
x
\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx
∫
e
x
−
e
−
x
e
x
+
e
−
x
d
x
[5]
∫
e
x
e
x
+
1
d
x
\int \frac{e^x}{e^x+1}dx
∫
e
x
+
1
e
x
d
x
2
1
Hide problems
Today's calculation of Integral 2
Calculate the following indefinite integrals. [1]
∫
cos
(
2
x
−
π
3
)
d
x
\int \cos \left(2x-\frac{\pi}{3}\right)dx
∫
cos
(
2
x
−
3
π
)
d
x
[2]
∫
d
x
cos
2
(
3
x
+
4
)
\int \frac{dx}{\cos ^2 (3x+4)}
∫
c
o
s
2
(
3
x
+
4
)
d
x
[3]
∫
(
x
−
1
)
x
−
2
3
d
x
\int (x-1)\sqrt[3]{x-2}dx
∫
(
x
−
1
)
3
x
−
2
d
x
[4]
∫
x
⋅
3
x
2
+
1
d
x
\int x\cdot 3^{x^2+1}dx
∫
x
⋅
3
x
2
+
1
d
x
[5]
∫
d
x
1
−
x
d
x
\int \frac{dx}{\sqrt{1-x}}dx
∫
1
−
x
d
x
d
x
1
1
Hide problems
Today's calculation of Integral 1
Calculate the following indefinite integral. [1]
∫
e
2
x
(
e
x
+
1
)
2
d
x
\int \frac{e^{2x}}{(e^x+1)^2}dx
∫
(
e
x
+
1
)
2
e
2
x
d
x
[2]
∫
sin
x
cos
3
x
d
x
\int \sin x\cos 3x dx
∫
sin
x
cos
3
x
d
x
[3]
∫
sin
2
x
sin
3
x
d
x
\int \sin 2x\sin 3x dx
∫
sin
2
x
sin
3
x
d
x
[4]
∫
d
x
4
x
2
−
12
x
+
9
\int \frac{dx}{4x^2-12x+9}
∫
4
x
2
−
12
x
+
9
d
x
[5]
∫
cos
4
x
d
x
\int \cos ^4 x dx
∫
cos
4
x
d
x