Question 1

Find the integral using an appropriate trigonometric substitution. $$\int \frac { \sqrt { 25 x ^ { 2 } - 16 } } { x } d x$$

Accepted Answer

The answer of Find the integral using an appropriate trigonometric...

Question 2

Evaluate the integral. $$\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x$$

Accepted Answer

The answer of Evaluate the integral. \[\int _ { 0...

Question 3

Determine whether the integral converges or diverges.If it converges, find its value. $$\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }$$

Accepted Answer

The answer of Determine whether the integral converges or diverges.If...

Question 4

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value.
$$\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4$$

Accepted Answer

a. @#LAT-DLM&
b. @#LAT-DLM&
Ex

Question 5

Evaluate the integral. $$\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$$

Accepted Answer

A)  $6 \arctan \left( \sec ^ { 2 } x \right) + C$ 
B)  $- 6 \cos ^ { 2 } 2 x + C$ 
C)  $- 6 \arcsin ( \tan x ) + C$ 
D)  $1 - 6 \cos ^ { 2 } x + C$ 
E)  None of these 
A)  $6 \arctan \left( \sec ^ { 2 } x \right) + C$ 
B)  $- 6 \cos ^ { 2 } 2 x + C$ 
C)  $- 6 \arcsin ( \tan x ) + C$ 
D)  $1 - 6 \cos ^ { 2 } x + C$ 
E)  None of these

Question 6

Evaluate the integral.
$$\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$$

Accepted Answer

A)  $\frac { 5 } { 32 } \pi$ 
B)  $\frac { 3 } { 16 } \pi$ 
C)  $\frac { 5 } { 16 } \pi$ 
D)  $\frac { 3 } { 32 } \pi$ 
A)  $\frac { 5 } { 32 } \pi$ 
B)  $\frac { 3 } { 16 } \pi$ 
C)  $\frac { 5 } { 16 } \pi$ 
D)  $\frac { 3 } { 32 } \pi$

Question 7

Evaluate the integral to six decimal places. $$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$$

Accepted Answer

The answer of Evaluate the integral to six decimal places....

Question 8

Evaluate the integral. $$\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  divergent 
A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  divergent

Question 9

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Accepted Answer

The answer of Make a substitution to express the integrand...

Question 10

Find the volume of the resulting solid if the region under the curve $y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }$ from $x = 0$ to $x = 1$ is rotated about the $x$-axis. Round your answer to four decimal places.

Accepted Answer

The answer of Find the volume of the resulting solid...

Question 11

Evaluate the integral. $$\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x$$

Accepted Answer

The answer of Evaluate the integral. \[\int \frac { 1...

Question 12

Evaluate the integral. $$\int _ { - 1 } ^ { 1 / \sqrt { 3 } } \frac { e ^ { \arctan y } } { 1 + y ^ { 2 } } d y$$

Accepted Answer

The answer of Evaluate the integral. \[\int _ { -...

Question 13

Use a table of integrals to evaluate the integral. $\int x \sqrt { 2 + 2 x } d x$

Accepted Answer

A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$

Question 14

Evaluate the integral. $$\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x$$

Accepted Answer

A)  $4 \left( \frac { 1 } { 3 } x - \frac { 1 } { 3 } \ln x \right) + C$ 
B)  $4 \left( \frac { 3 } { 2 } \ln ( x + 2 ) - \frac { 1 } { 2 } \ln x \right) + C$ 
C)  $- 4 \left( \frac { \sqrt { 2 } } { 2 } \arctan x \right) + C$ 
D)  $- 4 ( \ln x - x ) + C$ 
E)  None of these 
A)  $4 \left( \frac { 1 } { 3 } x - \frac { 1 } { 3 } \ln x \right) + C$ 
B)  $4 \left( \frac { 3 } { 2 } \ln ( x + 2 ) - \frac { 1 } { 2 } \ln x \right) + C$ 
C)  $- 4 \left( \frac { \sqrt { 2 } } { 2 } \arctan x \right) + C$ 
D)  $- 4 ( \ln x - x ) + C$ 
E)  None of these

Question 15

Evaluate the integral. $$2 \int _ { 0 } ^ { x / 2 } \sin ^ { 3 } \theta \cos ^ { 2 } \theta d \theta$$

Accepted Answer

The answer of Evaluate the integral. \[2 \int _ {...

Question 16

Find the integral. $$\int \frac { x ^ { 3 } + 2 x + 5 } { ( x + 1 ) \left( x ^ { 2 } + 1 \right) } d x$$

Accepted Answer

The answer of Find the integral. \[\int \frac { x...

Question 17

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$.
$$\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$$

Accepted Answer

A)  $6 \sin \theta + 6 \cos \theta + C$ 
B)  $6 \theta \cos \theta + 6 \sin \theta + C$ 
C)  $6 \theta \sin \theta + 6 \cos \theta + C$ 
D)  $6 \sin \theta - 6 \cos \theta + C$ 
E)  None of these 
A)  $6 \sin \theta + 6 \cos \theta + C$ 
B)  $6 \theta \cos \theta + 6 \sin \theta + C$ 
C)  $6 \theta \sin \theta + 6 \cos \theta + C$ 
D)  $6 \sin \theta - 6 \cos \theta + C$ 
E)  None of these

Question 18

Find the integral.
$$\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x$$

Accepted Answer

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

Question 19

A body moves along a coordinate line in such a way that its velocity at any time $t$, where $0 \leq t \leq 6$, is given by $v ( t ) = t \sqrt { 36 - t ^ { 2 } }$
Find its position function if it is initially located at the origin.

Accepted Answer

The answer of A body moves along a coordinate line...

Question 20

The region $\left\{ ( x + y ) \mid x \geq - 7,0 \leq y \leq e ^ { - x / 5 } \right\}$ is represented below. Find the area of this region to two decimal places.

Accepted Answer

A)  $20.28$ 
B)  $17.89$ 
C)  $16.08$ 
D)  $15.89$ 
E)  $15.87$ 
A)  $20.28$ 
B)  $17.89$ 
C)  $16.08$ 
D)  $15.89$ 
E)  $15.87$

Find the integral using an appropriate trigonometric substitution. $\int \frac { \sqrt { 25 x ^ { 2 } - 16 } } { x } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x$

Determine whether the integral converges or diverges.If it converges, find its value. $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value. $\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4$

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

Evaluate the integral to six decimal places. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Find the volume of the resulting solid if the region under the curve $y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }$ from $x = 0$ to $x = 1$ is rotated about the $x$ -axis. Round your answer to four decimal places.

Evaluate the integral. $\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x$

Evaluate the integral. $\int _ { - 1 } ^ { 1 / \sqrt { 3 } } \frac { e ^ { \arctan y } } { 1 + y ^ { 2 } } d y$

Use a table of integrals to evaluate the integral. $\int x \sqrt { 2 + 2 x } d x$

Evaluate the integral. $\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x$

Evaluate the integral. $2 \int _ { 0 } ^ { x / 2 } \sin ^ { 3 } \theta \cos ^ { 2 } \theta d \theta$

Find the integral. $\int \frac { x ^ { 3 } + 2 x + 5 } { ( x + 1 ) \left( x ^ { 2 } + 1 \right) } d x$

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$ . $\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$

Find the integral. $\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x$

A body moves along a coordinate line in such a way that its velocity at any time $t$ , where $0 \leq t \leq 6$ , is given by $v ( t ) = t \sqrt { 36 - t ^ { 2 } }$ Find its position function if it is initially located at the origin.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

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Vector Calculus

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Exam 7: Techniques of Integration

Find the integral using an appropriate trigonometric substitution. ∫25x2−16xdx\int \frac { \sqrt { 25 x ^ { 2 } - 16 } } { x } d x∫x25x2−16​​dx

Evaluate the integral. ∫01/212xcos⁡πxdx\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x∫01/2​12xcosπxdx

Determine whether the integral converges or diverges.If it converges, find its value. ∫1∞dxx5ln⁡x\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }∫1∞​x5lnxdx​

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value. ∫0x3sin⁡xdx;n=4\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4∫0x​3sinxdx;n=4

Evaluate the integral. ∫6sin⁡2x1+cos⁡4xdx\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x∫1+cos4x6sin2x​dx

Evaluate the integral. ∫0π/2cos⁡6xdx\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x∫0π/2​cos6xdx

Evaluate the integral to six decimal places. ∫01x364−x2dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x∫01​64−x2​x3​dx

Evaluate the integral. ∫1∞dxxln⁡x\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }∫1∞​xlnxdx​

Make a substitution to express the integrand as a rational function and then evaluate the integral. ∫425xx−100dx\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x∫425​x−100x​​dx Round the answer to four decimal places.

Find the volume of the resulting solid if the region under the curve y=1(x2+3x+2)y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }y=(x2+3x+2)1​ from x=0x = 0x=0 to x=1x = 1x=1 is rotated about the xxx -axis. Round your answer to four decimal places.

Evaluate the integral. ∫1+3ex1−exdx\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x∫1−ex1+3ex​dx

Evaluate the integral. ∫−11/3earctan⁡y1+y2dy\int _ { - 1 } ^ { 1 / \sqrt { 3 } } \frac { e ^ { \arctan y } } { 1 + y ^ { 2 } } d y∫−11/3​​1+y2earctany​dy

Use a table of integrals to evaluate the integral. ∫x2+2xdx\int x \sqrt { 2 + 2 x } d x∫x2+2x​dx

Evaluate the integral. ∫4(x−1x2+2x)dx\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x∫4(x2+2xx−1​)dx

Evaluate the integral. 2∫0x/2sin⁡3θcos⁡2θdθ2 \int _ { 0 } ^ { x / 2 } \sin ^ { 3 } \theta \cos ^ { 2 } \theta d \theta2∫0x/2​sin3θcos2θdθ

Find the integral. ∫x3+2x+5(x+1)(x2+1)dx\int \frac { x ^ { 3 } + 2 x + 5 } { ( x + 1 ) \left( x ^ { 2 } + 1 \right) } d x∫(x+1)(x2+1)x3+2x+5​dx

Evaluate the integral using integration by parts with the indicated choices of uuu and dvd vdv . ∫6θcos⁡θdθ,u=6θ,dv=cos⁡θdθ\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta∫6θcosθdθ,u=6θ,dv=cosθdθ

Find the integral. ∫3x−5x2−2x−3dx\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x∫x2−2x−33x−5​dx

A body moves along a coordinate line in such a way that its velocity at any time ttt , where 0≤t≤60 \leq t \leq 60≤t≤6 , is given by v(t)=t36−t2v ( t ) = t \sqrt { 36 - t ^ { 2 } }v(t)=t36−t2​ Find its position function if it is initially located at the origin.

The region {(x+y)∣x≥−7,0≤y≤e−x/5}\left\{ ( x + y ) \mid x \geq - 7,0 \leq y \leq e ^ { - x / 5 } \right\}{(x+y)∣x≥−7,0≤y≤e−x/5} is represented below. Find the area of this region to two decimal places.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Find the integral using an appropriate trigonometric substitution. $\int \frac { \sqrt { 25 x ^ { 2 } - 16 } } { x } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x$

Determine whether the integral converges or diverges.If it converges, find its value. $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value. $\int _ { 0 } ^ { x } 3 \sin x d x ; \quad n = 4$

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

Evaluate the integral to six decimal places. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Find the volume of the resulting solid if the region under the curve $y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }$ from $x = 0$ to $x = 1$ is rotated about the $x$ -axis. Round your answer to four decimal places.

Evaluate the integral. $\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x$

Evaluate the integral. $\int _ { - 1 } ^ { 1 / \sqrt { 3 } } \frac { e ^ { \arctan y } } { 1 + y ^ { 2 } } d y$

Use a table of integrals to evaluate the integral. $\int x \sqrt { 2 + 2 x } d x$

Evaluate the integral. $\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x$

Evaluate the integral. $2 \int _ { 0 } ^ { x / 2 } \sin ^ { 3 } \theta \cos ^ { 2 } \theta d \theta$

Find the integral. $\int \frac { x ^ { 3 } + 2 x + 5 } { ( x + 1 ) \left( x ^ { 2 } + 1 \right) } d x$

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$ . $\int 6 \theta \cos \theta d \theta , u = 6 \theta , d v = \cos \theta d \theta$

Find the integral. $\int \frac { 3 x - 5 } { x ^ { 2 } - 2 x - 3 } d x$

A body moves along a coordinate line in such a way that its velocity at any time $t$ , where $0 \leq t \leq 6$ , is given by $v ( t ) = t \sqrt { 36 - t ^ { 2 } }$ Find its position function if it is initially located at the origin.