Question 1

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

Accepted Answer

$\ln \left( 2 + e ^ { 2 } \right) - \ln 3$

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

$\frac { 1 } { 2 } x - \frac { 1 } { 4 } \sin 2 x + C$

Question 4

Find $\int \frac { x ^ { 2 } } { e ^ { 2 x } } d x$

Accepted Answer

A)  $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } + \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$ 
B)  $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } + \frac { 1 } { 4 } e ^ { - 2 x } + C$ 
C)  $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$ 
D)  $\frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$ 
A)  $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } + \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$ 
B)  $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } + \frac { 1 } { 4 } e ^ { - 2 x } + C$ 
C)  $- \frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$ 
D)  $\frac { 1 } { 2 } x ^ { 2 } e ^ { - 2 x } - \frac { 1 } { 2 } x e ^ { - 2 x } - \frac { 1 } { 4 } e ^ { - 2 x } + C$

Question 5

Find $\int \sec ( x ) \cos ^ { 4 } ( x ) d x$

Accepted Answer

The answer of Find \(\int \sec ( x ) \cos...

Question 6

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \tan ( x ) } { \cos ^ { 3 } ( x ) } d x$

Accepted Answer

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

Question 7

Evaluate $\int _ { 0 } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x$

Accepted Answer

A)  $1 / 2$ 
B)  $1 / 4$ 
C)  1/8 
D)  0 
A)  $1 / 2$ 
B)  $1 / 4$ 
C)  1/8 
D)  0

Question 8

Find $\int \frac { \sqrt { x } - 5 } { \sqrt { x } } d x$

Accepted Answer

A)  $x - \frac { 10 } { \sqrt { x } } + C$ 
B)  $x - 5 \sqrt { x } + C$ 
C)  $x + 10 \sqrt { x } + C$ 
D)  $x - 10 \sqrt { x } + C$ 
A)  $x - \frac { 10 } { \sqrt { x } } + C$ 
B)  $x - 5 \sqrt { x } + C$ 
C)  $x + 10 \sqrt { x } + C$ 
D)  $x - 10 \sqrt { x } + C$

Question 9

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

Accepted Answer

The answer of Find \(\int \sec ( x ) \tan...

Question 10

Find $\int \frac { 1 } { \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } } d x$

Accepted Answer

The answer of Find \(\int \frac { 1 } {...

Question 11

Find $\int x ^ { 4 } \sin \left( x ^ { 5 } \right) d x$

Accepted Answer

A)  $\frac { 1 } { 5 } \sin \left( x ^ { 5 } \right) + C$ 
B)  $\frac { 1 } { 5 } \cos \left( x ^ { 5 } \right) + C$ 
C)  $\frac { - 1 } { 5 } \cos \left( x ^ { 5 } \right) + C$ 
D)  $\frac { - 1 } { 5 } \sin \left( x ^ { 5 } \right) + C$ 
A)  $\frac { 1 } { 5 } \sin \left( x ^ { 5 } \right) + C$ 
B)  $\frac { 1 } { 5 } \cos \left( x ^ { 5 } \right) + C$ 
C)  $\frac { - 1 } { 5 } \cos \left( x ^ { 5 } \right) + C$ 
D)  $\frac { - 1 } { 5 } \sin \left( x ^ { 5 } \right) + C$

Question 12

Evaluate $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( x ) d x$

Accepted Answer

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

Question 13

Find $\int \sec ^ { 4 } ( x ) d x$

Accepted Answer

The answer of Find \(\int \sec ^ { 4 }...

Question 14

Find $\int x e ^ { 4 x } d x$

Accepted Answer

A)  $\frac { 1 } { 4 } x e ^ { 4 x } + \frac { 1 } { 16 } e ^ { 4 x } + C$ 
B)  $\frac { 1 } { 4 } x e ^ { 4 x } - \frac { 1 } { 4 } e ^ { 4 x } + C$ 
C)  $\frac { 1 } { 4 } x e ^ { 4 x } + \frac { 1 } { 4 } e ^ { 4 x } + C$ 
D)  $\frac { 1 } { 4 } x e ^ { 4 x } - \frac { 1 } { 16 } e ^ { 4 x } + C$ 
A)  $\frac { 1 } { 4 } x e ^ { 4 x } + \frac { 1 } { 16 } e ^ { 4 x } + C$ 
B)  $\frac { 1 } { 4 } x e ^ { 4 x } - \frac { 1 } { 4 } e ^ { 4 x } + C$ 
C)  $\frac { 1 } { 4 } x e ^ { 4 x } + \frac { 1 } { 4 } e ^ { 4 x } + C$ 
D)  $\frac { 1 } { 4 } x e ^ { 4 x } - \frac { 1 } { 16 } e ^ { 4 x } + C$

Question 15

Evaluate $\int _ { 0 } ^ { \pi 2 } e ^ { x } \cos ( x ) d x$

Accepted Answer

A)  1 
B)  0 
C)  $\frac { 1 } { 2 } e ^ { \pi / 2 }$ 
D)  $\frac { 1 } { 2 } e ^ { \pi / 2 } - \frac { 1 } { 2 }$ 
A)  1 
B)  0 
C)  $\frac { 1 } { 2 } e ^ { \pi / 2 }$ 
D)  $\frac { 1 } { 2 } e ^ { \pi / 2 } - \frac { 1 } { 2 }$

Question 16

Find $\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x$

Accepted Answer

The answer of Find \(\int \frac { \sqrt { 4...

Question 17

Find the integral using substitution and integration by parts $\int \frac { \cos ( x ) } { \sin ( x ) + \sin ^ { 2 } ( x ) } d x$

Accepted Answer

The answer of Find the integral using substitution and integration...

Question 18

Find $\int \cos ^ { 3 } ( x ) \tan ( x ) d x$

Accepted Answer

A)  $\frac { 1 } { 5 } \sin ^ { 5 } x + C$ 
B)  $\frac { 1 } { 3 } \sin ^ { 3 } x + C$ 
C)  $\frac { 1 } { 3 } \cos ^ { 3 } x + C$ 
D)  $- \frac { 1 } { 3 } \cos ^ { 3 } x + C$ 
A)  $\frac { 1 } { 5 } \sin ^ { 5 } x + C$ 
B)  $\frac { 1 } { 3 } \sin ^ { 3 } x + C$ 
C)  $\frac { 1 } { 3 } \cos ^ { 3 } x + C$ 
D)  $- \frac { 1 } { 3 } \cos ^ { 3 } x + C$

Question 19

Evaluate $\int _ { - 1 } ^ { 2 } \frac { 2 } { x ^ { 2 } - 9 } d x$

Accepted Answer

The answer of Evaluate \(\int _ { - 1 }...

Question 20

Evaluate $\int _ { 0 } ^ { \pi / 4 } \tan ( x ) \sec ^ { 2 } ( x ) d x$

Accepted Answer

A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 2 } } { 2 }$ 
C)  1 
D)  $1 / 2$ 
A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 2 } } { 2 }$ 
C)  1 
D)  $1 / 2$

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

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

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

Find $\int \frac { x ^ { 2 } } { e ^ { 2 x } } d x$

Find $\int \sec ( x ) \cos ^ { 4 } ( x ) d x$

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \tan ( x ) } { \cos ^ { 3 } ( x ) } d x$

Evaluate $\int _ { 0 } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x$

Find $\int \frac { \sqrt { x } - 5 } { \sqrt { x } } d x$

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

Find $\int \frac { 1 } { \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } } d x$

Find $\int x ^ { 4 } \sin \left( x ^ { 5 } \right) d x$

Evaluate $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( x ) d x$

Find $\int \sec ^ { 4 } ( x ) d x$

Find $\int x e ^ { 4 x } d x$

Evaluate $\int _ { 0 } ^ { \pi 2 } e ^ { x } \cos ( x ) d x$

Find $\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x$

Find the integral using substitution and integration by parts $\int \frac { \cos ( x ) } { \sin ( x ) + \sin ^ { 2 } ( x ) } d x$

Find $\int \cos ^ { 3 } ( x ) \tan ( x ) d x$

Evaluate $\int _ { - 1 } ^ { 2 } \frac { 2 } { x ^ { 2 } - 9 } d x$

Evaluate $\int _ { 0 } ^ { \pi / 4 } \tan ( x ) \sec ^ { 2 } ( x ) d x$

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

Evaluate ∫02ex2+exdx\int _ { 0 } ^ { 2 } \frac { e ^ { x } } { 2 + e ^ { x } } d x∫02​2+exex​dx

Find ∫01ln⁡(x2)xdx\int _ { 0 } ^ { 1 } \frac { \ln \left( x ^ { 2 } \right) } { x } d x∫01​xln(x2)​dx

Find ∫sin⁡3(x)csc⁡(x)dx\int \sin ^ { 3 } ( x ) \csc ( x ) d x∫sin3(x)csc(x)dx

Find ∫x2e2xdx\int \frac { x ^ { 2 } } { e ^ { 2 x } } d x∫e2xx2​dx

Find ∫sec⁡(x)cos⁡4(x)dx\int \sec ( x ) \cos ^ { 4 } ( x ) d x∫sec(x)cos4(x)dx

Evaluate ∫0π/3tan⁡(x)cos⁡3(x)dx\int _ { 0 } ^ { \pi / 3 } \frac { \tan ( x ) } { \cos ^ { 3 } ( x ) } d x∫0π/3​cos3(x)tan(x)​dx

Evaluate ∫0∞x(x2+4)2dx\int _ { 0 } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x∫0∞​(x2+4)2x​dx

Find ∫x−5xdx\int \frac { \sqrt { x } - 5 } { \sqrt { x } } d x∫x​x​−5​dx

Find ∫sec⁡(x)tan⁡5(x)dx\int \sec ( x ) \tan ^ { 5 } ( x ) d x∫sec(x)tan5(x)dx

Find ∫1(9−x2)3/2dx\int \frac { 1 } { \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } } d x∫(9−x2)3/21​dx

Find ∫x4sin⁡(x5)dx\int x ^ { 4 } \sin \left( x ^ { 5 } \right) d x∫x4sin(x5)dx

Evaluate ∫0π/4sin⁡2(x)dx\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( x ) d x∫0π/4​sin2(x)dx

Find ∫sec⁡4(x)dx\int \sec ^ { 4 } ( x ) d x∫sec4(x)dx

Find ∫xe4xdx\int x e ^ { 4 x } d x∫xe4xdx

Evaluate ∫0π2excos⁡(x)dx\int _ { 0 } ^ { \pi 2 } e ^ { x } \cos ( x ) d x∫0π2​excos(x)dx

Find ∫4−x2xdx\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x∫x4−x2​​dx

Find the integral using substitution and integration by parts ∫cos⁡(x)sin⁡(x)+sin⁡2(x)dx\int \frac { \cos ( x ) } { \sin ( x ) + \sin ^ { 2 } ( x ) } d x∫sin(x)+sin2(x)cos(x)​dx

Find ∫cos⁡3(x)tan⁡(x)dx\int \cos ^ { 3 } ( x ) \tan ( x ) d x∫cos3(x)tan(x)dx

Evaluate ∫−122x2−9dx\int _ { - 1 } ^ { 2 } \frac { 2 } { x ^ { 2 } - 9 } d x∫−12​x2−92​dx

Evaluate ∫0π/4tan⁡(x)sec⁡2(x)dx\int _ { 0 } ^ { \pi / 4 } \tan ( x ) \sec ^ { 2 } ( x ) d x∫0π/4​tan(x)sec2(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

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

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

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

Find $\int \frac { x ^ { 2 } } { e ^ { 2 x } } d x$

Find $\int \sec ( x ) \cos ^ { 4 } ( x ) d x$

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \tan ( x ) } { \cos ^ { 3 } ( x ) } d x$

Evaluate $\int _ { 0 } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x$

Find $\int \frac { \sqrt { x } - 5 } { \sqrt { x } } d x$

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

Find $\int \frac { 1 } { \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } } d x$

Find $\int x ^ { 4 } \sin \left( x ^ { 5 } \right) d x$

Evaluate $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( x ) d x$

Find $\int \sec ^ { 4 } ( x ) d x$

Find $\int x e ^ { 4 x } d x$

Evaluate $\int _ { 0 } ^ { \pi 2 } e ^ { x } \cos ( x ) d x$

Find $\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x$

Find the integral using substitution and integration by parts $\int \frac { \cos ( x ) } { \sin ( x ) + \sin ^ { 2 } ( x ) } d x$

Find $\int \cos ^ { 3 } ( x ) \tan ( x ) d x$

Evaluate $\int _ { - 1 } ^ { 2 } \frac { 2 } { x ^ { 2 } - 9 } d x$

Evaluate $\int _ { 0 } ^ { \pi / 4 } \tan ( x ) \sec ^ { 2 } ( x ) d x$