Question 1

Find $\int _ { 0 } ^ { 1 } \frac { 1 } { x ( x - 2 ) } d x$

Accepted Answer

A)  0 
B)  1 
C)  1/2 
D)  Divergent 
A)  0 
B)  1 
C)  1/2 
D)  Divergent

Question 2

Find $\int \sin ^ { 2 } ( x ) d x$

Accepted Answer

The answer of Find \(\int \sin ^ { 2 }...

Question 3

Evaluate $\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 2 } } { 1 + x ^ { 6 } } d x$

Accepted Answer

A)  $\pi$ /2 
B)  $\pi$ /4 
C)  3 $\pi$ /4 
D)  $\pi$ 
A)  $\pi$ /2 
B)  $\pi$ /4 
C)  3 $\pi$ /4 
D)  $\pi$

Question 4

Find $\int \sec ^ { 5 } ( x ) \tan ( x ) d x$

Accepted Answer

A)  $\frac { 1 } { 5 } \tan ^ { 5 } x + C$ 
B)  $\frac { 1 } { 5 } \sec ^ { 5 } x + C$ 
C)  $\frac { 1 } { 5 } \sec ^ { 5 } x \tan ^ { 5 } x + C$ 
D)  $\frac { 1 } { 3 } \sec ^ { 3 } x + C$ 
A)  $\frac { 1 } { 5 } \tan ^ { 5 } x + C$ 
B)  $\frac { 1 } { 5 } \sec ^ { 5 } x + C$ 
C)  $\frac { 1 } { 5 } \sec ^ { 5 } x \tan ^ { 5 } x + C$ 
D)  $\frac { 1 } { 3 } \sec ^ { 3 } x + C$

Question 5

Evaluate $\int _ { 0 }^?x e ^ { - 2 x } d x$

Accepted Answer

A)  1/2 
B)  1/4 
C)  0 
D)  1 
A)  1/2 
B)  1/4 
C)  0 
D)  1

Question 6

Find $\int \cos ( x ) e ^ { - 2 \sin ( x ) } d x$

Accepted Answer

A)  $\frac { 1 } { 2 } e ^ { - 2 \sin x } + C$ 
B)  $\frac { - 1 } { 2 } e ^ { - 2 \cos x } + C$ 
C)  $- e ^ { - 2 \sin x } + C$ 
D)  $\frac { - 1 } { 2 } e ^ { - 2 \sin x } + C$ 
A)  $\frac { 1 } { 2 } e ^ { - 2 \sin x } + C$ 
B)  $\frac { - 1 } { 2 } e ^ { - 2 \cos x } + C$ 
C)  $- e ^ { - 2 \sin x } + C$ 
D)  $\frac { - 1 } { 2 } e ^ { - 2 \sin x } + C$

Question 7

Find $\int x ^ { 3 } \cos \left( x ^ { 2 } \right) d x$

Accepted Answer

A)  $\frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) - \cos \left( x ^ { 2 } \right) + C$ 
B)  $\frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + C$ 
C)  $- \frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + C$ 
D)  $- \frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) - \cos \left( x ^ { 2 } \right) + C$ 
A)  $\frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) - \cos \left( x ^ { 2 } \right) + C$ 
B)  $\frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + C$ 
C)  $- \frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + C$ 
D)  $- \frac { 1 } { 2 } x ^ { 2 } \sin \left( x ^ { 2 } \right) - \cos \left( x ^ { 2 } \right) + C$

Question 8

Find $\int x \ln 3 x d x$

Accepted Answer

The answer of Find \(\int x \ln 3 x d...

Question 9

Find $\int e ^ { x } \ln \left( e ^ { 3 x } \right) d x$

Accepted Answer

The answer of Find \(\int e ^ { x }...

Question 10

Evaluate $\int _ { 0 } ^ { \infty } \frac { 1 } { 1 + x ^ { 2 } } d x$

Accepted Answer

The answer of Evaluate \(\int _ { 0 } ^...

Question 11

Find $\int x \cos ( 3 x ) d x$

Accepted Answer

A)  $\frac { 1 } { 3 } x \sin 3 x - \frac { 1 } { 9 } \cos 3 x + C$ 
B)  $\frac { 1 } { 3 } x \cos 3 x + \frac { 1 } { 9 } \sin 3 x + C$ 
C)  $- \frac { 1 } { 3 } x \sin 3 x + \frac { 1 } { 9 } \cos 3 x + C$ 
D)  $\frac { 1 } { 3 } x \sin 3 x + \frac { 1 } { 9 } \cos 3 x + C$ 
A)  $\frac { 1 } { 3 } x \sin 3 x - \frac { 1 } { 9 } \cos 3 x + C$ 
B)  $\frac { 1 } { 3 } x \cos 3 x + \frac { 1 } { 9 } \sin 3 x + C$ 
C)  $- \frac { 1 } { 3 } x \sin 3 x + \frac { 1 } { 9 } \cos 3 x + C$ 
D)  $\frac { 1 } { 3 } x \sin 3 x + \frac { 1 } { 9 } \cos 3 x + C$

Question 12

Find $\int \sec ^ { 6 } ( x ) \tan ( x ) d x$

Accepted Answer

The answer of Find \(\int \sec ^ { 6 }...

Question 13

Find $\int x \left( x ^ { 2 } - 1 \right) ^ { 4 } d x$

Accepted Answer

A)  $\frac { 1 } { 8 } \left( x ^ { 2 } - 1 \right) ^ { 4 } + C$ 
B)  $\frac { 1 } { 8 } \left( x ^ { 2 } + 1 \right) ^ { 4 } + C$ 
C)  $\frac { 1 } { 10 } \left( x ^ { 2 } - 1 \right) ^ { 5 } + C$ 
D)  $\frac { 1 } { 10 } \left( x ^ { 2 } + 1 \right) ^ { 5 } + C$ 
A)  $\frac { 1 } { 8 } \left( x ^ { 2 } - 1 \right) ^ { 4 } + C$ 
B)  $\frac { 1 } { 8 } \left( x ^ { 2 } + 1 \right) ^ { 4 } + C$ 
C)  $\frac { 1 } { 10 } \left( x ^ { 2 } - 1 \right) ^ { 5 } + C$ 
D)  $\frac { 1 } { 10 } \left( x ^ { 2 } + 1 \right) ^ { 5 } + C$

Question 14

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $$\tan \left( \sin ^ { - 1 } ( x ) \right)$$

Accepted Answer

The answer of Use reference triangles and unit circle to...

Question 15

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 2 \right) \sqrt { x } d x$

Accepted Answer

A)  12/7 
B)  14/5 
C)  14/9 
D)  12/9 
A)  12/7 
B)  14/5 
C)  14/9 
D)  12/9

Question 16

Find $\int _ { 1 } ^ { e } \frac { 1 } { x \ln x } d x$

Accepted Answer

A)  0 
B)  1 
C)  1/2 
D)  Divergent 
A)  0 
B)  1 
C)  1/2 
D)  Divergent

Question 17

Find $\int \sec ( 3 x ) \tan ( 3 x ) d x$

Accepted Answer

A)  $\sec 3 x + C$ 
B)  $\frac { 1 } { 2 } \sec ^ { 2 } 3 x + C$ 
C)  $\frac { 1 } { 3 } \sec 3 x \tan 3 x + C$ 
D)  $\frac { 1 } { 3 } \sec 3 x + C$ 
A)  $\sec 3 x + C$ 
B)  $\frac { 1 } { 2 } \sec ^ { 2 } 3 x + C$ 
C)  $\frac { 1 } { 3 } \sec 3 x \tan 3 x + C$ 
D)  $\frac { 1 } { 3 } \sec 3 x + C$

Question 18

Find $\int \frac { x ^ { 3 } } { x ^ { 2 } + 1 } d x$

Accepted Answer

The answer of Find \(\int \frac { x ^ {...

Question 19

Find $\int \sin ^ { - 1 } ( 3 x ) d x$

Accepted Answer

The answer of Find \(\int \sin ^ { - 1...

Question 20

Evaluate $\int _ { - \pi } ^ { \pi } \sin ^ { 3 } ( x ) d x$

Accepted Answer

A)  1 
B)  0 
C)  $1 / 2$ 
D)  -1 
A)  1 
B)  0 
C)  $1 / 2$ 
D)  -1

Find $\int _ { 0 } ^ { 1 } \frac { 1 } { x ( x - 2 ) } d x$

Find $\int \sin ^ { 2 } ( x ) d x$

Evaluate $\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 2 } } { 1 + x ^ { 6 } } d x$

Find $\int \sec ^ { 5 } ( x ) \tan ( x ) d x$

Evaluate $\int _ { 0 }^?x e ^ { - 2 x } d x$

Find $\int \cos ( x ) e ^ { - 2 \sin ( x ) } d x$

Find $\int x ^ { 3 } \cos \left( x ^ { 2 } \right) d x$

Find $\int x \ln 3 x d x$

Find $\int e ^ { x } \ln \left( e ^ { 3 x } \right) d x$

Evaluate $\int _ { 0 } ^ { \infty } \frac { 1 } { 1 + x ^ { 2 } } d x$

Find $\int x \cos ( 3 x ) d x$

Find $\int \sec ^ { 6 } ( x ) \tan ( x ) d x$

Find $\int x \left( x ^ { 2 } - 1 \right) ^ { 4 } d x$

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $\tan \left( \sin ^ { - 1 } ( x ) \right)$

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 2 \right) \sqrt { x } d x$

Find $\int _ { 1 } ^ { e } \frac { 1 } { x \ln x } d x$

Find $\int \sec ( 3 x ) \tan ( 3 x ) d x$

Find $\int \frac { x ^ { 3 } } { x ^ { 2 } + 1 } d x$

Find $\int \sin ^ { - 1 } ( 3 x ) d x$

Evaluate $\int _ { - \pi } ^ { \pi } \sin ^ { 3 } ( x ) d x$

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 5: Techniques of Integration

Find ∫011x(x−2)dx\int _ { 0 } ^ { 1 } \frac { 1 } { x ( x - 2 ) } d x∫01​x(x−2)1​dx

Find ∫sin⁡2(x)dx\int \sin ^ { 2 } ( x ) d x∫sin2(x)dx

Evaluate ∫013x21+x6dx\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 2 } } { 1 + x ^ { 6 } } d x∫01​1+x63x2​dx

Find ∫sec⁡5(x)tan⁡(x)dx\int \sec ^ { 5 } ( x ) \tan ( x ) d x∫sec5(x)tan(x)dx

Evaluate ∫0?xe−2xdx\int _ { 0 }^?x e ^ { - 2 x } d x∫0?​xe−2xdx

Find ∫cos⁡(x)e−2sin⁡(x)dx\int \cos ( x ) e ^ { - 2 \sin ( x ) } d x∫cos(x)e−2sin(x)dx

Find ∫x3cos⁡(x2)dx\int x ^ { 3 } \cos \left( x ^ { 2 } \right) d x∫x3cos(x2)dx

Find ∫xln⁡3xdx\int x \ln 3 x d x∫xln3xdx

Find ∫exln⁡(e3x)dx\int e ^ { x } \ln \left( e ^ { 3 x } \right) d x∫exln(e3x)dx

Evaluate ∫0∞11+x2dx\int _ { 0 } ^ { \infty } \frac { 1 } { 1 + x ^ { 2 } } d x∫0∞​1+x21​dx

Find ∫xcos⁡(3x)dx\int x \cos ( 3 x ) d x∫xcos(3x)dx

Find ∫sec⁡6(x)tan⁡(x)dx\int \sec ^ { 6 } ( x ) \tan ( x ) d x∫sec6(x)tan(x)dx

Find ∫x(x2−1)4dx\int x \left( x ^ { 2 } - 1 \right) ^ { 4 } d x∫x(x2−1)4dx

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: tan⁡(sin⁡−1(x))\tan \left( \sin ^ { - 1 } ( x ) \right)tan(sin−1(x))

Evaluate ∫01(x3+2)xdx\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 2 \right) \sqrt { x } d x∫01​(x3+2)x​dx

Find ∫1e1xln⁡xdx\int _ { 1 } ^ { e } \frac { 1 } { x \ln x } d x∫1e​xlnx1​dx

Find ∫sec⁡(3x)tan⁡(3x)dx\int \sec ( 3 x ) \tan ( 3 x ) d x∫sec(3x)tan(3x)dx

Find ∫x3x2+1dx\int \frac { x ^ { 3 } } { x ^ { 2 } + 1 } d x∫x2+1x3​dx

Find ∫sin⁡−1(3x)dx\int \sin ^ { - 1 } ( 3 x ) d x∫sin−1(3x)dx

Evaluate ∫−ππsin⁡3(x)dx\int _ { - \pi } ^ { \pi } \sin ^ { 3 } ( x ) d x∫−ππ​sin3(x)dx

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Find $\int _ { 0 } ^ { 1 } \frac { 1 } { x ( x - 2 ) } d x$

Find $\int \sin ^ { 2 } ( x ) d x$

Evaluate $\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 2 } } { 1 + x ^ { 6 } } d x$

Find $\int \sec ^ { 5 } ( x ) \tan ( x ) d x$

Evaluate $\int _ { 0 }^?x e ^ { - 2 x } d x$

Find $\int \cos ( x ) e ^ { - 2 \sin ( x ) } d x$

Find $\int x ^ { 3 } \cos \left( x ^ { 2 } \right) d x$

Find $\int x \ln 3 x d x$

Find $\int e ^ { x } \ln \left( e ^ { 3 x } \right) d x$

Evaluate $\int _ { 0 } ^ { \infty } \frac { 1 } { 1 + x ^ { 2 } } d x$

Find $\int x \cos ( 3 x ) d x$

Find $\int \sec ^ { 6 } ( x ) \tan ( x ) d x$

Find $\int x \left( x ^ { 2 } - 1 \right) ^ { 4 } d x$

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $\tan \left( \sin ^ { - 1 } ( x ) \right)$

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 2 \right) \sqrt { x } d x$

Find $\int _ { 1 } ^ { e } \frac { 1 } { x \ln x } d x$

Find $\int \sec ( 3 x ) \tan ( 3 x ) d x$

Find $\int \frac { x ^ { 3 } } { x ^ { 2 } + 1 } d x$

Find $\int \sin ^ { - 1 } ( 3 x ) d x$

Evaluate $\int _ { - \pi } ^ { \pi } \sin ^ { 3 } ( x ) d x$