Question 1

Find $\int \frac { e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } } d x$

Accepted Answer

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

Question 2

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

Accepted Answer

Question 3

Find $\int \sqrt { 9 - x ^ { 2 } } d x$

Accepted Answer

The answer of Find \(\int \sqrt { 9 - x...

Question 4

Find $\int \frac { \sqrt [ 3 ] { \ln x } } { x } d x$

Accepted Answer

A)  $\frac { 1 } { 4 } ( \ln x ) ^ { 1 / 3 } + C$ 
B)  $\frac { 3 } { 4 } ( \ln x ) ^ { 4 / 3 } + C$ 
C)  $( \ln x ) ^ { 1 / 3 } + C$ 
D)  $\ln x + C$ 
A)  $\frac { 1 } { 4 } ( \ln x ) ^ { 1 / 3 } + C$ 
B)  $\frac { 3 } { 4 } ( \ln x ) ^ { 4 / 3 } + C$ 
C)  $( \ln x ) ^ { 1 / 3 } + C$ 
D)  $\ln x + C$

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Find $\int \frac { 5 } { x ^ { 2 } - 1 } d x$

Accepted Answer

The answer of Find \(\int \frac { 5 } {...

Question 8

Find $\int \tan ^ { - 1 } ( 2 x ) d x$

Accepted Answer

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

Question 9

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

Accepted Answer

The answer of Find \(\int \sin ^ { 3 }...

Question 10

Find $\int \sin ^ { 5 } ( x ) \cot ( x ) d x$

Accepted Answer

A)  $\frac { 1 } { 5 } \sec ^ { 5 } x + C$ 
B)  $\frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
C)  $\frac { 1 } { 5 } \sin ^ { 5 } x + C$ 
D)  $\frac { 1 } { 6 } \sin ^ { 6 } x + C$ 
A)  $\frac { 1 } { 5 } \sec ^ { 5 } x + C$ 
B)  $\frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
C)  $\frac { 1 } { 5 } \sin ^ { 5 } x + C$ 
D)  $\frac { 1 } { 6 } \sin ^ { 6 } x + C$

Question 11

Evaluate $\int _ { 3 } ^ { \infty } \frac { 1 } { x ( x - 2 ) } d x$

Accepted Answer

The answer of Evaluate \(\int _ { 3 } ^...

Question 12

Find $\int \frac { 1 } { x ( \ln x ) ^ { 3 } } d x$

Accepted Answer

A)  $\frac { 1 } { ( \ln x ) ^ { 2 } } + C$ 
B)  $\frac { 1 } { 3 ( \ln x ) ^ { 3 } } + C$ 
C)  $\frac { - 1 } { 2 ( \ln x ) ^ { 2 } } + C$ 
D)  $\frac { 1 } { 2 ( \ln x ) ^ { 2 } } + C$ 
A)  $\frac { 1 } { ( \ln x ) ^ { 2 } } + C$ 
B)  $\frac { 1 } { 3 ( \ln x ) ^ { 3 } } + C$ 
C)  $\frac { - 1 } { 2 ( \ln x ) ^ { 2 } } + C$ 
D)  $\frac { 1 } { 2 ( \ln x ) ^ { 2 } } + C$

Question 13

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

Accepted Answer

Question 14

Find $\int \ln ( 5 x ) d x$

Accepted Answer

A)  $x \ln 5 x - x + C$ 
B)  $x \ln 5 x + x + C$ 
C)  $x \ln x - x + C$ 
D)  $5 x \ln x - x + C$ 
A)  $x \ln 5 x - x + C$ 
B)  $x \ln 5 x + x + C$ 
C)  $x \ln x - x + C$ 
D)  $5 x \ln x - x + C$

Question 15

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

Accepted Answer

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

Question 16

Evaluate $\int _ { 2 } ^ { 3 } \frac { 3 } { x ( x - 1 ) } d x$

Accepted Answer

A)  $4 \ln \left( \frac { 4 } { 3 } \right)$ 
B)  $3 \ln \left( \frac { 3 } { 4 } \right)$ 
C)  $3 \ln \left( \frac { 4 } { 3 } \right)$ 
D)  $4 \ln \left( \frac { 3 } { 4 } \right)$ 
A)  $4 \ln \left( \frac { 4 } { 3 } \right)$ 
B)  $3 \ln \left( \frac { 3 } { 4 } \right)$ 
C)  $3 \ln \left( \frac { 4 } { 3 } \right)$ 
D)  $4 \ln \left( \frac { 3 } { 4 } \right)$

Question 17

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

Accepted Answer

A)  $2 x ( \ln x + 1 ) + C$ 
B)  $2 x \ln x - x + C$ 
C)  $2 x ( \ln x - 1 ) + C$ 
D)  $2 x \ln x + x + C$ 
A)  $2 x ( \ln x + 1 ) + C$ 
B)  $2 x \ln x - x + C$ 
C)  $2 x ( \ln x - 1 ) + C$ 
D)  $2 x \ln x + x + C$

Question 18

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

Accepted Answer

The answer of Find \(\int \cos ^ { 3 }...

Question 19

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \sin ( x ) } { \cos ^ { 2 } ( x ) } d x$

Accepted Answer

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

Question 20

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

Accepted Answer

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

Find $\int \frac { e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } } d x$

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

Find $\int \sqrt { 9 - x ^ { 2 } } d x$

Find $\int \frac { \sqrt [ 3 ] { \ln x } } { x } d x$

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

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

Find $\int \frac { 5 } { x ^ { 2 } - 1 } d x$

Find $\int \tan ^ { - 1 } ( 2 x ) d x$

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

Find $\int \sin ^ { 5 } ( x ) \cot ( x ) d x$

Evaluate $\int _ { 3 } ^ { \infty } \frac { 1 } { x ( x - 2 ) } d x$

Find $\int \frac { 1 } { x ( \ln x ) ^ { 3 } } d x$

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

Find $\int \ln ( 5 x ) d x$

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

Evaluate $\int _ { 2 } ^ { 3 } \frac { 3 } { x ( x - 1 ) } d x$

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

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

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \sin ( x ) } { \cos ^ { 2 } ( x ) } d x$

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

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

Find ∫e2x1−e4xdx\int \frac { e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } } d x∫1−e4x​e2x​dx

Find ∫xsec⁡2(x)dx\int x \sec ^ { 2 } ( x ) d x∫xsec2(x)dx

Find ∫9−x2dx\int \sqrt { 9 - x ^ { 2 } } d x∫9−x2​dx

Find ∫ln⁡x3xdx\int \frac { \sqrt [ 3 ] { \ln x } } { x } d x∫x3lnx​​dx

Find ∫x2−4xdx\int \frac { \sqrt { x ^ { 2 } - 4 } } { x } d x∫xx2−4​​dx

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

Find ∫5x2−1dx\int \frac { 5 } { x ^ { 2 } - 1 } d x∫x2−15​dx

Find ∫tan⁡−1(2x)dx\int \tan ^ { - 1 } ( 2 x ) d x∫tan−1(2x)dx

Find ∫sin⁡3(2x)cos⁡2(2x)dx\int \sin ^ { 3 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x∫sin3(2x)cos2(2x)dx

Find ∫sin⁡5(x)cot⁡(x)dx\int \sin ^ { 5 } ( x ) \cot ( x ) d x∫sin5(x)cot(x)dx

Evaluate ∫3∞1x(x−2)dx\int _ { 3 } ^ { \infty } \frac { 1 } { x ( x - 2 ) } d x∫3∞​x(x−2)1​dx

Find ∫1x(ln⁡x)3dx\int \frac { 1 } { x ( \ln x ) ^ { 3 } } d x∫x(lnx)31​dx

Find ∫tan⁡(x)ln⁡(ecos⁡(x))dx\int \tan ( x ) \ln \left( e ^ { \cos ( x ) } \right) d x∫tan(x)ln(ecos(x))dx

Find ∫ln⁡(5x)dx\int \ln ( 5 x ) d x∫ln(5x)dx

Find ∫x2sin⁡(x)dx\int x ^ { 2 } \sin ( x ) d x∫x2sin(x)dx

Evaluate ∫233x(x−1)dx\int _ { 2 } ^ { 3 } \frac { 3 } { x ( x - 1 ) } d x∫23​x(x−1)3​dx

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

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

Evaluate ∫0π/3sin⁡(x)cos⁡2(x)dx\int _ { 0 } ^ { \pi / 3 } \frac { \sin ( x ) } { \cos ^ { 2 } ( x ) } d x∫0π/3​cos2(x)sin(x)​dx

Find ∫cos⁡2(2x)dx\int \cos ^ { 2 } ( 2 x ) d x∫cos2(2x)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 $\int \frac { e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } } d x$

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

Find $\int \sqrt { 9 - x ^ { 2 } } d x$

Find $\int \frac { \sqrt [ 3 ] { \ln x } } { x } d x$

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

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

Find $\int \frac { 5 } { x ^ { 2 } - 1 } d x$

Find $\int \tan ^ { - 1 } ( 2 x ) d x$

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

Find $\int \sin ^ { 5 } ( x ) \cot ( x ) d x$

Evaluate $\int _ { 3 } ^ { \infty } \frac { 1 } { x ( x - 2 ) } d x$

Find $\int \frac { 1 } { x ( \ln x ) ^ { 3 } } d x$

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

Find $\int \ln ( 5 x ) d x$

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

Evaluate $\int _ { 2 } ^ { 3 } \frac { 3 } { x ( x - 1 ) } d x$

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

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

Evaluate $\int _ { 0 } ^ { \pi / 3 } \frac { \sin ( x ) } { \cos ^ { 2 } ( x ) } d x$

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