Question 1

Use various trigonometric identities to simplify the expression then integrate.
-$$\int \cos ^ { 7 } \theta \sin 2 \theta d \theta$$

Accepted Answer

A)  $\frac { 1 } { 4 } \cos ^ { 7 } \theta + C$ 
B)  $- \frac { 1 } { 5 } \cos ^ { 10 } \theta + C$ 
C)  $\frac { 1 } { 9 } \cos ^ { 8 } \theta + C$ 
D)  $- \frac { 2 } { 9 } \cos ^ { 9 } \theta + C$ 
A)  $\frac { 1 } { 4 } \cos ^ { 7 } \theta + C$ 
B)  $- \frac { 1 } { 5 } \cos ^ { 10 } \theta + C$ 
C)  $\frac { 1 } { 9 } \cos ^ { 8 } \theta + C$ 
D)  $- \frac { 2 } { 9 } \cos ^ { 9 } \theta + C$

Question 2

Evaluate the improper integral.
-$$\int _ { 0 } ^ { 25 } \frac { d x } { \sqrt { | x - 16 | } }$$

Accepted Answer

A)  6 
B)  7 
C)  14 
D)  -2 
A)  6 
B)  7 
C)  14 
D)  -2

Question 3

Find the value of the constant k so that the given function in a probability density function for a random variable over the
specified interval.
-$f ( x ) = k ^ { x }$ over $[ 0,20 ]$

Accepted Answer

A)  $\frac { 1 } { \mathrm { e } ^ { 20 } - 1 }$ 
B)  $\frac { 1 } { \mathrm { e } ^ { 20 } + 1 }$ 
C)  $\mathrm { e } ^ { 20 } - 1$ 
D)  $\frac { 2 } { \mathrm { e } ^ { 20 } }$ 
A)  $\frac { 1 } { \mathrm { e } ^ { 20 } - 1 }$ 
B)  $\frac { 1 } { \mathrm { e } ^ { 20 } + 1 }$ 
C)  $\mathrm { e } ^ { 20 } - 1$ 
D)  $\frac { 2 } { \mathrm { e } ^ { 20 } }$

Question 4

Solve the problem by integration.
-The force $\mathrm { F }$ (in N) applied by a stamping machine in making a certain computer part is $\mathrm { F } = \frac { 3 \mathrm { x } } { \mathrm { x } ^ { 2 } + 11 \mathrm { x } + 12 }$, where $\mathrm { x }$ is the distance (in $\mathrm { cm }$ ) through which the force acts. Find the work done by the force from $\mathrm { x } = 0$ to $\mathrm { x } = 0.6 \mathrm {~cm}$.

Accepted Answer

A)  0.0832 N· cm 
B)  8.0928 N· cm 
C)  0.0166 N· cm 
D)  0.0055 N· cm 
A)  0.0832 N· cm 
B)  8.0928 N· cm 
C)  0.0166 N· cm 
D)  0.0055 N· cm

Question 5

Evaluate the integral.
-$\int \frac { 1 } { x \ln x ^ { 6 } } d x$

Accepted Answer

A)  $\ln \left( \ln x ^ { 6 } \right) + C$ 
B)  $\ln x ^ { 6 } + C$ 
C)  $\frac { 1 } { 6 } \ln x ^ { 6 } + C$ 
D)  $\frac { 1 } { 6 } \ln \left( \ln x ^ { 6 } \right) + C$ 
A)  $\ln \left( \ln x ^ { 6 } \right) + C$ 
B)  $\ln x ^ { 6 } + C$ 
C)  $\frac { 1 } { 6 } \ln x ^ { 6 } + C$ 
D)  $\frac { 1 } { 6 } \ln \left( \ln x ^ { 6 } \right) + C$

Question 6

Solve the initial value problem for y as a function of x.
-$$\sqrt { x ^ { 2 } - 4 } \frac { d y } { d x } = x , x > 2 , y ( 4 ) = 0$$

Accepted Answer

A)  $y = \sqrt { x ^ { 2 } - 4 } - 2 \sqrt { 3 }$ 
B)  $y = \sqrt { x ^ { 2 } - 4 }$ 
C)  $y = \frac { \sqrt { x ^ { 2 } - 4 } } { x }$ 
D)  $y = \frac { \sqrt { x ^ { 2 } - 4 } } { x } - \frac { \sqrt { 3 } } { 2 }$ 
A)  $y = \sqrt { x ^ { 2 } - 4 } - 2 \sqrt { 3 }$ 
B)  $y = \sqrt { x ^ { 2 } - 4 }$ 
C)  $y = \frac { \sqrt { x ^ { 2 } - 4 } } { x }$ 
D)  $y = \frac { \sqrt { x ^ { 2 } - 4 } } { x } - \frac { \sqrt { 3 } } { 2 }$

Question 7

Solve the problem.
-Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.14 onces and a standard
Deviation of 0.04 ounce. Find the probability that the bottle contains more than 12.14 ounces of beer.

Accepted Answer

A)  0.4 
B)  1 
C)  0 
D)  0.5 
A)  0.4 
B)  1 
C)  0 
D)  0.5

Question 8

Provide an appropriate response.
-$\text { The "Simpson" sum is based upon the area under a } \underline { ? } \text {. }$

Accepted Answer

A)  rectangle 
B)  trapezoid 
C)  triangle 
D)  parabola 
A)  rectangle 
B)  trapezoid 
C)  triangle 
D)  parabola

Question 9

Solve the initial value problem for y as a function of x.
-$$x \frac { d y } { d x } = \sqrt { x ^ { 2 } - 9 } , x \geq 3 , y ( 3 ) = 0$$

Accepted Answer

A)  $y = 3 \ln \frac { x } { 3 }$ 
B)  $y = \frac { \sqrt { x ^ { 2 } - 9 } } { 3 } - x + 3$ 
C)  $y = \frac { \sqrt { x ^ { 2 } - 9 } } { 3 \sec ^ { - 1 } ( x / 3 ) }$ 
D)  $y = 3 \left[ \frac { \sqrt { \mathrm { x } ^ { 2 } - 9 } } { 3 } - \sec ^ { - 1 } \left( \frac { \mathrm { x } } { 3 } \right) \right]$ 
A)  $y = 3 \ln \frac { x } { 3 }$ 
B)  $y = \frac { \sqrt { x ^ { 2 } - 9 } } { 3 } - x + 3$ 
C)  $y = \frac { \sqrt { x ^ { 2 } - 9 } } { 3 \sec ^ { - 1 } ( x / 3 ) }$ 
D)  $y = 3 \left[ \frac { \sqrt { \mathrm { x } ^ { 2 } - 9 } } { 3 } - \sec ^ { - 1 } \left( \frac { \mathrm { x } } { 3 } \right) \right]$

Question 10

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula.
-$$\int \frac { d x } { \left( 25 - x ^ { 2 } \right) ^ { 2 } }$$

Accepted Answer

A)  $\frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| + C$ 
B)  $\frac { x } { 50 \left( 25 - x ^ { 2 } \right) } + C$ 
C)  $\frac { 1 } { 50 } \left( \frac { x } { 25 - x ^ { 2 } } + \frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| \right) + C$ 
D)  $\frac { 1 } { 50 } \left( \frac { x } { 25 - x ^ { 2 } } - \frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| \right) + C$ 
A)  $\frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| + C$ 
B)  $\frac { x } { 50 \left( 25 - x ^ { 2 } \right) } + C$ 
C)  $\frac { 1 } { 50 } \left( \frac { x } { 25 - x ^ { 2 } } + \frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| \right) + C$ 
D)  $\frac { 1 } { 50 } \left( \frac { x } { 25 - x ^ { 2 } } - \frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| \right) + C$

Question 11

Determine whether the improper integral converges or diverges.
-$$\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 2 / 5 } + 2 }$$

Accepted Answer

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

Question 12

Use reduction formulas to evaluate the integral.
-$\int \sin ^ { 2 } 3 x \cos ^ { 2 } 3 x d x$

Accepted Answer

A)  $- \frac { 1 } { 3 } \sin 3 x \cos ^ { 3 } 3 x + \frac { x } { 8 } + \frac { 1 } { 3 } \sin 6 x + C$ 
B)  $- \frac { 1 } { 12 } \sin 3 x \cos ^ { 2 } 3 x + \frac { x } { 8 } + \frac { 1 } { 48 } \sin 6 x + C$ 
C)  $- \frac { 1 } { 12 } \sin 3 x \cos ^ { 3 } 3 x + \frac { x } { 8 } + \frac { 1 } { 48 } \sin 6 x + C$ 
D)  $- \frac { 1 } { 12 } \sin 3 x \cos ^ { 3 } 3 x + \frac { x } { 8 } + \frac { 1 } { 48 } \cos 6 x + C$ 
A)  $- \frac { 1 } { 3 } \sin 3 x \cos ^ { 3 } 3 x + \frac { x } { 8 } + \frac { 1 } { 3 } \sin 6 x + C$ 
B)  $- \frac { 1 } { 12 } \sin 3 x \cos ^ { 2 } 3 x + \frac { x } { 8 } + \frac { 1 } { 48 } \sin 6 x + C$ 
C)  $- \frac { 1 } { 12 } \sin 3 x \cos ^ { 3 } 3 x + \frac { x } { 8 } + \frac { 1 } { 48 } \sin 6 x + C$ 
D)  $- \frac { 1 } { 12 } \sin 3 x \cos ^ { 3 } 3 x + \frac { x } { 8 } + \frac { 1 } { 48 } \cos 6 x + C$

Question 13

Use reduction formulas to evaluate the integral.
-$$\int 8 \cos ^ { 3 } 2 x d x$$

Accepted Answer

A)  $4 \sin 2 x - \frac { 4 } { 3 } \cos ^ { 3 } 2 x + C$ 
B)  $4 \sin 2 x + \frac { 4 } { 3 } \sin ^ { 3 } 2 x + C$ 
C)  $8 \sin 2 x - \frac { 8 } { 3 } \sin ^ { 3 } 2 x + C$ 
D)  $4 \sin 2 x - \frac { 4 } { 3 } \sin ^ { 3 } 2 x + C$ 
A)  $4 \sin 2 x - \frac { 4 } { 3 } \cos ^ { 3 } 2 x + C$ 
B)  $4 \sin 2 x + \frac { 4 } { 3 } \sin ^ { 3 } 2 x + C$ 
C)  $8 \sin 2 x - \frac { 8 } { 3 } \sin ^ { 3 } 2 x + C$ 
D)  $4 \sin 2 x - \frac { 4 } { 3 } \sin ^ { 3 } 2 x + C$

Question 14

Expand the quotient by partial fractions.
-$$\frac { x + 4 } { x ^ { 2 } + 2 x + 1 }$$

Accepted Answer

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

Question 15

Solve the problem.
-Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 1 } ^ { 5 } \left( 3 \mathrm { x } ^ { 2 } + 8 \right) \mathrm { dx }$ with $\mathrm { n } = 6$ steps.

Accepted Answer

A)  $\frac { 125 } { 72 }$ 
B)  $\frac { 8 } { 9 }$ 
C)  $\frac { 4 } { 9 }$ 
D)  $\frac { 4 } { 27 }$ 
A)  $\frac { 125 } { 72 }$ 
B)  $\frac { 8 } { 9 }$ 
C)  $\frac { 4 } { 9 }$ 
D)  $\frac { 4 } { 27 }$

Question 16

Provide an appropriate response.
-(a) Find the values of $p$ for which $\int _ { 0 } ^ { 1 } \frac { 1 } { x p } d x$ converges.
(b) Find the values of $p$ for which $\int _ { 0 } ^ { 1 } \frac { 1 } { x _ { p } } d x$ diverges.

Accepted Answer

(a) The integral con

Question 17

Integrate the function.
-$$\int \frac { d x } { \sqrt { 25 x ^ { 2 } - 121 } } , x > \frac { 11 } { 5 }$$

Accepted Answer

A)  $\frac { 1 } { 11 } \ln \left| \frac { 11 } { 5 } x + \frac { \sqrt { 25 x ^ { 2 } - 121 } } { 5 } \right| + C$ 
B)  $\frac { 1 } { 5 } \ln \left| \sec ^ { - 1 } \left( \frac { 11 } { 5 } x \right) + \frac { \sqrt { 25 x ^ { 2 } - 121 } } { 11 } \right| + C$ 
C)  $\frac { 1 } { 5 } \ln \left| \frac { 5 } { 11 } x + \frac { \sqrt { 25 x ^ { 2 } - 121 } } { 11 } \right| + C$ 
D)  $\frac { 1 } { 5 } \ln \left| \frac { 5 } { 2 } x + \frac { 11 } { \sqrt { 25 x ^ { 2 } - 121 } } \right| + C$ 
A)  $\frac { 1 } { 11 } \ln \left| \frac { 11 } { 5 } x + \frac { \sqrt { 25 x ^ { 2 } - 121 } } { 5 } \right| + C$ 
B)  $\frac { 1 } { 5 } \ln \left| \sec ^ { - 1 } \left( \frac { 11 } { 5 } x \right) + \frac { \sqrt { 25 x ^ { 2 } - 121 } } { 11 } \right| + C$ 
C)  $\frac { 1 } { 5 } \ln \left| \frac { 5 } { 11 } x + \frac { \sqrt { 25 x ^ { 2 } - 121 } } { 11 } \right| + C$ 
D)  $\frac { 1 } { 5 } \ln \left| \frac { 5 } { 2 } x + \frac { 11 } { \sqrt { 25 x ^ { 2 } - 121 } } \right| + C$

Question 18

Evaluate the integral.
-Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int x \left( \frac { d } { d x } f ^ { - 1 } ( x ) \right) d x$ to evaluate the integral. $\int \cot ^ { - 1 } x d x$

Accepted Answer

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

Question 19

Use Simpson's Rule with n = 4 steps to estimate the integral.
-$$\int _ { - 1 } ^ { 0 } \sin \pi \mathrm { tdt }$$

Accepted Answer

A)  $- \frac { 1 + \sqrt { 2 } } { 4 }$ 
B)  $- 1 - 2 \sqrt { 2 }$ 
C)  $- \frac { 1 + 2 \sqrt { 2 } } { 6 }$ 
D)  $- \frac { 2 + \sqrt { 2 } } { 6 }$ 
A)  $- \frac { 1 + \sqrt { 2 } } { 4 }$ 
B)  $- 1 - 2 \sqrt { 2 }$ 
C)  $- \frac { 1 + 2 \sqrt { 2 } } { 6 }$ 
D)  $- \frac { 2 + \sqrt { 2 } } { 6 }$

Question 20

Determine whether the improper integral converges or diverges.
-$$\int _ { - \infty } ^ { \infty } \frac { d x } { \sqrt { 5 x ^ { 6 } + 1 } }$$

Accepted Answer

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

Use various trigonometric identities to simplify the expression then integrate. - $\int \cos ^ { 7 } \theta \sin 2 \theta d \theta$

Evaluate the improper integral. - $\int _ { 0 } ^ { 25 } \frac { d x } { \sqrt { | x - 16 | } }$

Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. - $f ( x ) = k ^ { x }$ over $[ 0,20 ]$

Evaluate the integral. - $\int \frac { 1 } { x \ln x ^ { 6 } } d x$

Solve the initial value problem for y as a function of x. - $\sqrt { x ^ { 2 } - 4 } \frac { d y } { d x } = x , x > 2 , y ( 4 ) = 0$

Provide an appropriate response. - $\text { The "Simpson" sum is based upon the area under a } \underline { ? } \text {. }$

Solve the initial value problem for y as a function of x. - $x \frac { d y } { d x } = \sqrt { x ^ { 2 } - 9 } , x \geq 3 , y ( 3 ) = 0$

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. - $\int \frac { d x } { \left( 25 - x ^ { 2 } \right) ^ { 2 } }$

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 2 / 5 } + 2 }$

Use reduction formulas to evaluate the integral. - $\int \sin ^ { 2 } 3 x \cos ^ { 2 } 3 x d x$

Use reduction formulas to evaluate the integral. - $\int 8 \cos ^ { 3 } 2 x d x$

Expand the quotient by partial fractions. - $\frac { x + 4 } { x ^ { 2 } + 2 x + 1 }$

Solve the problem. -Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 1 } ^ { 5 } \left( 3 \mathrm { x } ^ { 2 } + 8 \right) \mathrm { dx }$ with $\mathrm { n } = 6$ steps.

Provide an appropriate response. -(a) Find the values of $p$ for which $\int _ { 0 } ^ { 1 } \frac { 1 } { x p } d x$ converges. (b) Find the values of $p$ for which $\int _ { 0 } ^ { 1 } \frac { 1 } { x _ { p } } d x$ diverges.

Integrate the function. - $\int \frac { d x } { \sqrt { 25 x ^ { 2 } - 121 } } , x > \frac { 11 } { 5 }$

Evaluate the integral. -Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int x \left( \frac { d } { d x } f ^ { - 1 } ( x ) \right) d x$ to evaluate the integral. $\int \cot ^ { - 1 } x d x$

Use Simpson's Rule with n = 4 steps to estimate the integral. - $\int _ { - 1 } ^ { 0 } \sin \pi \mathrm { tdt }$

Determine whether the improper integral converges or diverges. - $\int _ { - \infty } ^ { \infty } \frac { d x } { \sqrt { 5 x ^ { 6 } + 1 } }$

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Exam 9: Techniques of Integration

Use various trigonometric identities to simplify the expression then integrate. - ∫cos⁡7θsin⁡2θdθ\int \cos ^ { 7 } \theta \sin 2 \theta d \theta∫cos7θsin2θdθ

Evaluate the improper integral. - ∫025dx∣x−16∣\int _ { 0 } ^ { 25 } \frac { d x } { \sqrt { | x - 16 | } }∫025​∣x−16∣​dx​

Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. - f(x)=kxf ( x ) = k ^ { x }f(x)=kx over [0,20][ 0,20 ][0,20]

Evaluate the integral. - ∫1xln⁡x6dx\int \frac { 1 } { x \ln x ^ { 6 } } d x∫xlnx61​dx

Solve the initial value problem for y as a function of x. - x2−4dydx=x,x>2,y(4)=0\sqrt { x ^ { 2 } - 4 } \frac { d y } { d x } = x , x > 2 , y ( 4 ) = 0x2−4​dxdy​=x,x>2,y(4)=0

Provide an appropriate response. - The "Simpson" sum is based upon the area under a ?‾. \text { The "Simpson" sum is based upon the area under a } \underline { ? } \text {. } The "Simpson" sum is based upon the area under a ?​.

Solve the initial value problem for y as a function of x. - xdydx=x2−9,x≥3,y(3)=0x \frac { d y } { d x } = \sqrt { x ^ { 2 } - 9 } , x \geq 3 , y ( 3 ) = 0xdxdy​=x2−9​,x≥3,y(3)=0

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. - ∫dx(25−x2)2\int \frac { d x } { \left( 25 - x ^ { 2 } \right) ^ { 2 } }∫(25−x2)2dx​

Determine whether the improper integral converges or diverges. - ∫1∞dxx2/5+2\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 2 / 5 } + 2 }∫1∞​x2/5+2dx​

Use reduction formulas to evaluate the integral. - ∫sin⁡23xcos⁡23xdx\int \sin ^ { 2 } 3 x \cos ^ { 2 } 3 x d x∫sin23xcos23xdx

Use reduction formulas to evaluate the integral. - ∫8cos⁡32xdx\int 8 \cos ^ { 3 } 2 x d x∫8cos32xdx

Expand the quotient by partial fractions. - x+4x2+2x+1\frac { x + 4 } { x ^ { 2 } + 2 x + 1 }x2+2x+1x+4​

Solve the problem. -Find an upper bound for ∣ET∣\left| \mathrm { E } _ { \mathrm { T } } \right|∣ET​∣ in estimating ∫15(3x2+8)dx\int _ { 1 } ^ { 5 } \left( 3 \mathrm { x } ^ { 2 } + 8 \right) \mathrm { dx }∫15​(3x2+8)dx with n=6\mathrm { n } = 6n=6 steps.

Provide an appropriate response. -(a) Find the values of ppp for which ∫011xpdx\int _ { 0 } ^ { 1 } \frac { 1 } { x p } d x∫01​xp1​dx converges. (b) Find the values of ppp for which ∫011xpdx\int _ { 0 } ^ { 1 } \frac { 1 } { x _ { p } } d x∫01​xp​1​dx diverges.

Integrate the function. - ∫dx25x2−121,x>115\int \frac { d x } { \sqrt { 25 x ^ { 2 } - 121 } } , x > \frac { 11 } { 5 }∫25x2−121​dx​,x>511​

Use Simpson's Rule with n = 4 steps to estimate the integral. - ∫−10sin⁡πtdt\int _ { - 1 } ^ { 0 } \sin \pi \mathrm { tdt }∫−10​sinπtdt

Determine whether the improper integral converges or diverges. - ∫−∞∞dx5x6+1\int _ { - \infty } ^ { \infty } \frac { d x } { \sqrt { 5 x ^ { 6 } + 1 } }∫−∞∞​5x6+1​dx​