Question 1

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 1 } ^ { \infty } \frac { 9 } { \left( 1 + x ^ { 2 } \right) \tan ^ { - 1 } x } d x$$

Accepted Answer

A)  $\frac { 9 } { 2 } \left( 1 + \frac { \pi } { 2 } \right) ^ { 2 }$ 
B)  $9 \ln 2$ 
C)  $9 \ln \frac { \pi } { 2 }$ 
D)  $9 \ln \left( 1 + \frac { \pi } { 2 } \right)$ 
A)  $\frac { 9 } { 2 } \left( 1 + \frac { \pi } { 2 } \right) ^ { 2 }$ 
B)  $9 \ln 2$ 
C)  $9 \ln \frac { \pi } { 2 }$ 
D)  $9 \ln \left( 1 + \frac { \pi } { 2 } \right)$

Question 2

Evaluate the integral.
-$$\int \frac { ( x + 5 ) ^ { 2 } \tan ^ { - 1 } ( 2 x ) + 4 x ^ { 3 } + x } { \left( 4 x ^ { 2 } + 1 \right) ( x + 5 ) ^ { 2 } } d x$$

Accepted Answer

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

Question 3

Solve the initial value problem for y as a function of x.
-$$\left( x ^ { 2 } + 81 \right) ^ { 2 } \frac { d y } { d x } = \sqrt { x ^ { 2 } + 81 } , y ( 0 ) = 9$$

Accepted Answer

A)  $y = \frac { x } { 9 } + 9$ 
B)  $y = \frac { x } { 81 } + 9$ 
C)  $y = \frac { x } { 81 \sqrt { x ^ { 2 } + 81 } } + 9$ 
D)  $y = \frac { x } { 81 \sqrt { x ^ { 2 } + 81 } }$ 
A)  $y = \frac { x } { 9 } + 9$ 
B)  $y = \frac { x } { 81 } + 9$ 
C)  $y = \frac { x } { 81 \sqrt { x ^ { 2 } + 81 } } + 9$ 
D)  $y = \frac { x } { 81 \sqrt { x ^ { 2 } + 81 } }$

Question 4

Find the area or volume.
-Find the area under the curve $y = \frac { 1 } { ( x + 1 ) ^ { 3 / 2 } }$ bounded on the left by $x = 15$.

Accepted Answer

A)  $\frac { 2 } { 15 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  8 
D)  $\frac { 1 } { 4 }$ 
A)  $\frac { 2 } { 15 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  8 
D)  $\frac { 1 } { 4 }$

Question 5

Use any method to evaluate the integral.
-$$\int \frac { 9 \csc ^ { 3 } x } { \tan x } d x$$

Accepted Answer

A)  $- 3 \csc ^ { 3 } x + C$ 
B)  $- 3 \csc ^ { 4 } x + C$ 
C)  $\frac { 9 } { 4 } \csc ^ { 4 } x \cot x + C$ 
D)  $- 3 \cot ^ { 3 } x + C$ 
A)  $- 3 \csc ^ { 3 } x + C$ 
B)  $- 3 \csc ^ { 4 } x + C$ 
C)  $\frac { 9 } { 4 } \csc ^ { 4 } x \cot x + C$ 
D)  $- 3 \cot ^ { 3 } x + C$

Question 6

Evaluate the integral.
-$\int _ { 0 } ^ { \pi / 2 } \cos 7 t \cos 6 t d t$

Accepted Answer

A)  $\frac { 9 } { 11 }$ 
B)  $\frac { 7 } { 13 }$ 
C)  $\frac { 7 } { 11 }$ 
D)  $\frac { 6 } { 13 }$ 
A)  $\frac { 9 } { 11 }$ 
B)  $\frac { 7 } { 13 }$ 
C)  $\frac { 7 } { 11 }$ 
D)  $\frac { 6 } { 13 }$

Question 7

Evaluate the integral.
-$$\int \cos ^ { 4 } \theta \sin 2 \theta d \theta$$

Accepted Answer

A)  $\frac { 2 } { 5 } \cos ^ { 4 } \theta + \mathrm { C }$ 
B)  $- \frac { 2 } { 7 } \cos ^ { 7 } \theta + \mathrm { C }$ 
C)  $\frac { 1 } { 6 } \cos ^ { 5 } \theta + C$ 
D)  $- \frac { 1 } { 3 } \cos ^ { 6 } \theta + C$ 
A)  $\frac { 2 } { 5 } \cos ^ { 4 } \theta + \mathrm { C }$ 
B)  $- \frac { 2 } { 7 } \cos ^ { 7 } \theta + \mathrm { C }$ 
C)  $\frac { 1 } { 6 } \cos ^ { 5 } \theta + C$ 
D)  $- \frac { 1 } { 3 } \cos ^ { 6 } \theta + C$

Question 8

Expand the quotient by partial fractions.
-$$\frac { x + 5 } { ( x + 2 ) ^ { 2 } }$$

Accepted Answer

A)  $\frac { 3 } { x + 2 } + \frac { 1 } { ( x + 2 ) ^ { 2 } }$ 
B)  $\frac { 1 } { x + 2 } + \frac { 4 } { x + 5 }$ 
C)  $\frac { 1 } { x + 2 } + \frac { - 3 } { ( x + 2 ) ^ { 2 } }$ 
D)  $\frac { 1 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } }$ 
A)  $\frac { 3 } { x + 2 } + \frac { 1 } { ( x + 2 ) ^ { 2 } }$ 
B)  $\frac { 1 } { x + 2 } + \frac { 4 } { x + 5 }$ 
C)  $\frac { 1 } { x + 2 } + \frac { - 3 } { ( x + 2 ) ^ { 2 } }$ 
D)  $\frac { 1 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } }$

Question 9

Evaluate the improper integral.
-$$\int _ { 0 } ^ { 2 } \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x$$

Accepted Answer

A)  -4 
B)  -2 
C)  4 
D)  2 
A)  -4 
B)  -2 
C)  4 
D)  2

Question 10

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions.
-$$\int \frac { 9 x ^ { 3 } + 18 x ^ { 2 } + 12 x + 5 } { 9 x ^ { 2 } + 18 x + 9 } d x$$

Accepted Answer

A)  $\frac { 1 } { 2 } x ^ { 2 } + \ln | 3 x + 3 | - \frac { 2 } { 3 x + 3 } + C$ 
B)  $\frac { 1 } { 2 } x ^ { 2 } - \frac { 3 x + 5 } { 3 x + 3 } + C$ 
C)  $\frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } \ln | 3 x + 3 | - \frac { 2 } { 3 } \frac { 1 } { 3 x + 3 } + C$ 
D)  $\frac { 1 } { 2 } x ^ { 2 } + \ln | 3 x + 3 | - \frac { 2 } { 3 } \frac { 1 } { 3 x + 3 } + C$ 
A)  $\frac { 1 } { 2 } x ^ { 2 } + \ln | 3 x + 3 | - \frac { 2 } { 3 x + 3 } + C$ 
B)  $\frac { 1 } { 2 } x ^ { 2 } - \frac { 3 x + 5 } { 3 x + 3 } + C$ 
C)  $\frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } \ln | 3 x + 3 | - \frac { 2 } { 3 } \frac { 1 } { 3 x + 3 } + C$ 
D)  $\frac { 1 } { 2 } x ^ { 2 } + \ln | 3 x + 3 | - \frac { 2 } { 3 } \frac { 1 } { 3 x + 3 } + C$

Question 11

Use any method to evaluate the integral.
-$$\int \frac { \cos x } { \sin ^ { 2 } x } d x$$

Accepted Answer

A)  sec x + C 
B)  csc x + C 
C)  -csc x + C 
D)  -sec x + C 
A)  sec x + C 
B)  csc x + C 
C)  -csc x + C 
D)  -sec x + C

Question 12

Solve the problem.
-Find the volume of the solid generated by revolving the region bounded by the curve $y = \ln x$, the $x$-axis, and the vertical line $x = e ^ { 2 }$ about the $x$-axis.

Accepted Answer

A)  $\pi \left( e ^ { 2 } - 1 \right)$ 
B)  $\pi ( e - 1 )$ 
C)  $2 \pi \left( \mathrm { e } ^ { 2 } - 1 \right)$ 
D)  $\pi \mathrm { e }$ 
A)  $\pi \left( e ^ { 2 } - 1 \right)$ 
B)  $\pi ( e - 1 )$ 
C)  $2 \pi \left( \mathrm { e } ^ { 2 } - 1 \right)$ 
D)  $\pi \mathrm { e }$

Question 13

Determine whether the improper integral converges or diverges.
-$$\int _ { 10 } ^ { \infty } \frac { d x } { \sqrt [ 6 ] { x - 9 } }$$

Accepted Answer

A)  Converges 
B)  Diverges 
A)  Converges 
B)  Diverges

Question 14

Evaluate the integral.
-$$\int _ { - \pi / 27 } ^ { \pi / 27 } \sec ^ { 3 } 9 x d x$$
Give your answer in exact form.

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 7 + 4 \sqrt { 3 } )$ 
B)  $\frac { 2 \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 2 \sqrt { 3 } )$ 
C)  $\frac { 2 } { 9 } + \frac { 1 } { 18 } \ln 4$ 
D)  $\frac { 2 \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 7 + 4 \sqrt { 3 } )$ 
A)  $\frac { \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 7 + 4 \sqrt { 3 } )$ 
B)  $\frac { 2 \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 2 \sqrt { 3 } )$ 
C)  $\frac { 2 } { 9 } + \frac { 1 } { 18 } \ln 4$ 
D)  $\frac { 2 \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 7 + 4 \sqrt { 3 } )$

Question 15

Evaluate the integral.
-$\int 3 \sin 5 x \sin 2 x d x$

Accepted Answer

A)  $3 \left( \frac { \cos 3 x } { 6 } - \frac { \cos 7 x } { 14 } \right) + C$ 
B)  $3 \left( \frac { \sin 3 x } { 6 } - \frac { \sin 7 x } { 14 } \right) + C$ 
C)  $3 \left( \frac { \sin 3 x } { 6 } + \frac { \sin 7 x } { 14 } \right) + C$ 
D)  $3 ( \sin 3 x - \sin 7 x ) + C$ 
A)  $3 \left( \frac { \cos 3 x } { 6 } - \frac { \cos 7 x } { 14 } \right) + C$ 
B)  $3 \left( \frac { \sin 3 x } { 6 } - \frac { \sin 7 x } { 14 } \right) + C$ 
C)  $3 \left( \frac { \sin 3 x } { 6 } + \frac { \sin 7 x } { 14 } \right) + C$ 
D)  $3 ( \sin 3 x - \sin 7 x ) + C$

Question 16

Evaluate the improper integral.
-$$\int _ { 0 } ^ { 4 } x \ln 4 x d x$$

Accepted Answer

A)  $8 \ln 16 - 4$ 
B)  $- \frac { 1 } { 2 } \ln 16 + 4$ 
C)  $8 \ln 4 - 4$ 
D)  $8 \ln 16 - 8$ 
A)  $8 \ln 16 - 4$ 
B)  $- \frac { 1 } { 2 } \ln 16 + 4$ 
C)  $8 \ln 4 - 4$ 
D)  $8 \ln 16 - 8$

Question 17

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int _ { 3 } ^ { 7 } \frac { 4 x ^ { 3 } - 4 x } { x ^ { 4 } - 16 } d x$$

Accepted Answer

A)  -3.4 
B)  3.4 
C)  3.8 
D)  6.81 
A)  -3.4 
B)  3.4 
C)  3.8 
D)  6.81

Question 18

Solve the problem.
-Two dice are tossed, and the random variable X assigns to each outcome the sum of the number of dots showing on each face. What is the probability that X = 5?

Accepted Answer

A)  4 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 8 } { 9 }$ 
D)  $\frac { 5 } { 6 }$ 
A)  4 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 8 } { 9 }$ 
D)  $\frac { 5 } { 6 }$

Question 19

Use a trigonometric substitution to evaluate the integral.
-$$\int _ { 0 } ^ { \ln 3 } \frac { e ^ { x } d x } { \sqrt { e ^ { 2 x } + 1 } }$$

Accepted Answer

A)  $\ln \left( \mathrm { e } ^ { 3 } + \sqrt { 10 } \right)$ 
B)  $\ln ( 3 + \sqrt { 10 } ) - \ln ( 1 + \sqrt { 2 } )$ 
C)  $\ln \frac { 3 } { 2 }$ 
D)  $\ln 6 - \ln ( 1 + \sqrt { 2 } )$ 
A)  $\ln \left( \mathrm { e } ^ { 3 } + \sqrt { 10 } \right)$ 
B)  $\ln ( 3 + \sqrt { 10 } ) - \ln ( 1 + \sqrt { 2 } )$ 
C)  $\ln \frac { 3 } { 2 }$ 
D)  $\ln 6 - \ln ( 1 + \sqrt { 2 } )$

Question 20

Evaluate the integral.
-$$\int _ { - \pi / 20 } ^ { \pi / 20 } \tan ^ { 4 } 5 \mathrm { tdt }$$

Accepted Answer

A)  $\frac { \pi } { 10 }$ 
B)  $\frac { \pi } { 10 } - \frac { 4 } { 15 }$ 
C)  $\frac { \pi } { 15 } - \frac { 2 } { 15 }$ 
D)  $- \frac { 4 } { 15 }$ 
A)  $\frac { \pi } { 10 }$ 
B)  $\frac { \pi } { 10 } - \frac { 4 } { 15 }$ 
C)  $\frac { \pi } { 15 } - \frac { 2 } { 15 }$ 
D)  $- \frac { 4 } { 15 }$

Evaluate the improper integral or state that it is divergent. - $\int _ { 1 } ^ { \infty } \frac { 9 } { \left( 1 + x ^ { 2 } \right) \tan ^ { - 1 } x } d x$

Evaluate the integral. - $\int \frac { ( x + 5 ) ^ { 2 } \tan ^ { - 1 } ( 2 x ) + 4 x ^ { 3 } + x } { \left( 4 x ^ { 2 } + 1 \right) ( x + 5 ) ^ { 2 } } d x$

Solve the initial value problem for y as a function of x. - $\left( x ^ { 2 } + 81 \right) ^ { 2 } \frac { d y } { d x } = \sqrt { x ^ { 2 } + 81 } , y ( 0 ) = 9$

Find the area or volume. -Find the area under the curve $y = \frac { 1 } { ( x + 1 ) ^ { 3 / 2 } }$ bounded on the left by $x = 15$ .

Use any method to evaluate the integral. - $\int \frac { 9 \csc ^ { 3 } x } { \tan x } d x$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 2 } \cos 7 t \cos 6 t d t$

Evaluate the integral. - $\int \cos ^ { 4 } \theta \sin 2 \theta d \theta$

Expand the quotient by partial fractions. - $\frac { x + 5 } { ( x + 2 ) ^ { 2 } }$

Evaluate the improper integral. - $\int _ { 0 } ^ { 2 } \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x$

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - $\int \frac { 9 x ^ { 3 } + 18 x ^ { 2 } + 12 x + 5 } { 9 x ^ { 2 } + 18 x + 9 } d x$

Use any method to evaluate the integral. - $\int \frac { \cos x } { \sin ^ { 2 } x } d x$

Solve the problem. -Find the volume of the solid generated by revolving the region bounded by the curve $y = \ln x$ , the $x$ -axis, and the vertical line $x = e ^ { 2 }$ about the $x$ -axis.

Determine whether the improper integral converges or diverges. - $\int _ { 10 } ^ { \infty } \frac { d x } { \sqrt [ 6 ] { x - 9 } }$

Evaluate the integral. - $\int _ { - \pi / 27 } ^ { \pi / 27 } \sec ^ { 3 } 9 x d x$ Give your answer in exact form.

Evaluate the integral. - $\int 3 \sin 5 x \sin 2 x d x$

Evaluate the improper integral. - $\int _ { 0 } ^ { 4 } x \ln 4 x d x$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int _ { 3 } ^ { 7 } \frac { 4 x ^ { 3 } - 4 x } { x ^ { 4 } - 16 } d x$

Solve the problem. -Two dice are tossed, and the random variable X assigns to each outcome the sum of the number of dots showing on each face. What is the probability that X = 5?

Use a trigonometric substitution to evaluate the integral. - $\int _ { 0 } ^ { \ln 3 } \frac { e ^ { x } d x } { \sqrt { e ^ { 2 x } + 1 } }$

Evaluate the integral. - $\int _ { - \pi / 20 } ^ { \pi / 20 } \tan ^ { 4 } 5 \mathrm { tdt }$

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Exam 9: Techniques of Integration

Evaluate the improper integral or state that it is divergent. - ∫1∞9(1+x2)tan⁡−1xdx\int _ { 1 } ^ { \infty } \frac { 9 } { \left( 1 + x ^ { 2 } \right) \tan ^ { - 1 } x } d x∫1∞​(1+x2)tan−1x9​dx

Evaluate the integral. - ∫(x+5)2tan⁡−1(2x)+4x3+x(4x2+1)(x+5)2dx\int \frac { ( x + 5 ) ^ { 2 } \tan ^ { - 1 } ( 2 x ) + 4 x ^ { 3 } + x } { \left( 4 x ^ { 2 } + 1 \right) ( x + 5 ) ^ { 2 } } d x∫(4x2+1)(x+5)2(x+5)2tan−1(2x)+4x3+x​dx

Solve the initial value problem for y as a function of x. - (x2+81)2dydx=x2+81,y(0)=9\left( x ^ { 2 } + 81 \right) ^ { 2 } \frac { d y } { d x } = \sqrt { x ^ { 2 } + 81 } , y ( 0 ) = 9(x2+81)2dxdy​=x2+81​,y(0)=9

Find the area or volume. -Find the area under the curve y=1(x+1)3/2y = \frac { 1 } { ( x + 1 ) ^ { 3 / 2 } }y=(x+1)3/21​ bounded on the left by x=15x = 15x=15 .

Use any method to evaluate the integral. - ∫9csc⁡3xtan⁡xdx\int \frac { 9 \csc ^ { 3 } x } { \tan x } d x∫tanx9csc3x​dx

Evaluate the integral. - ∫0π/2cos⁡7tcos⁡6tdt\int _ { 0 } ^ { \pi / 2 } \cos 7 t \cos 6 t d t∫0π/2​cos7tcos6tdt

Evaluate the integral. - ∫cos⁡4θsin⁡2θdθ\int \cos ^ { 4 } \theta \sin 2 \theta d \theta∫cos4θsin2θdθ

Expand the quotient by partial fractions. - x+5(x+2)2\frac { x + 5 } { ( x + 2 ) ^ { 2 } }(x+2)2x+5​

Evaluate the improper integral. - ∫02x4−x2dx\int _ { 0 } ^ { 2 } \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x∫02​4−x2​x​dx

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - ∫9x3+18x2+12x+59x2+18x+9dx\int \frac { 9 x ^ { 3 } + 18 x ^ { 2 } + 12 x + 5 } { 9 x ^ { 2 } + 18 x + 9 } d x∫9x2+18x+99x3+18x2+12x+5​dx

Use any method to evaluate the integral. - ∫cos⁡xsin⁡2xdx\int \frac { \cos x } { \sin ^ { 2 } x } d x∫sin2xcosx​dx

Solve the problem. -Find the volume of the solid generated by revolving the region bounded by the curve y=ln⁡xy = \ln xy=lnx , the xxx -axis, and the vertical line x=e2x = e ^ { 2 }x=e2 about the xxx -axis.

Determine whether the improper integral converges or diverges. - ∫10∞dxx−96\int _ { 10 } ^ { \infty } \frac { d x } { \sqrt [ 6 ] { x - 9 } }∫10∞​6x−9​dx​

Evaluate the integral. - ∫−π/27π/27sec⁡39xdx\int _ { - \pi / 27 } ^ { \pi / 27 } \sec ^ { 3 } 9 x d x∫−π/27π/27​sec39xdx Give your answer in exact form.

Evaluate the integral. - ∫3sin⁡5xsin⁡2xdx\int 3 \sin 5 x \sin 2 x d x∫3sin5xsin2xdx

Evaluate the improper integral. - ∫04xln⁡4xdx\int _ { 0 } ^ { 4 } x \ln 4 x d x∫04​xln4xdx

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫374x3−4xx4−16dx\int _ { 3 } ^ { 7 } \frac { 4 x ^ { 3 } - 4 x } { x ^ { 4 } - 16 } d x∫37​x4−164x3−4x​dx