Question 1

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$.
$$f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$$

Accepted Answer

A)  $\pi$ 
B)  0 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { \pi } { 12 }$ 
E)  $\frac { \pi } { 6 }$ 
A)  $\pi$ 
B)  0 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { \pi } { 12 }$ 
E)  $\frac { \pi } { 6 }$ B

Question 2

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places.
$$\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$$

Accepted Answer

0.2029

Question 3

Use the Midpoint Rule to approximate the given integral with the specified value of $n$. Compare your result to the actual value. Find the error in the approximation.
$$2 \int _ { 2 } ^ { 3 } e ^ { - \sqrt { x } } d x , n = 6$$

Accepted Answer

0.00008

Question 4

Evaluate the integral.
$$\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }$$

Accepted Answer

A)  $\frac { 1 } { 2 } \left( \tan ( x + 1 ) + \frac { 17 } { x ^ { 2 } + 2 x + 2 } \right) + C$ 
B)  $\frac { 1 } { 2 } \left( \tan ^ { - 1 } ( x + 2 ) + \frac { 7 } { x ^ { 2 } + 2 } \right) + C$ 
C)  $\frac { 1 } { 2 } \left( \tan ^ { - 1 } ( x + 7 ) + \frac { 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$ 
D)  $\frac { 7 } { 2 } \left( \tan ( x + 2 ) + \frac { x + 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$ 
E)  $\frac { 7 } { 2 } \left( \tan ^ { - 1 } ( x + 1 ) + \frac { x + 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$ 
A)  $\frac { 1 } { 2 } \left( \tan ( x + 1 ) + \frac { 17 } { x ^ { 2 } + 2 x + 2 } \right) + C$ 
B)  $\frac { 1 } { 2 } \left( \tan ^ { - 1 } ( x + 2 ) + \frac { 7 } { x ^ { 2 } + 2 } \right) + C$ 
C)  $\frac { 1 } { 2 } \left( \tan ^ { - 1 } ( x + 7 ) + \frac { 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$ 
D)  $\frac { 7 } { 2 } \left( \tan ( x + 2 ) + \frac { x + 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$ 
E)  $\frac { 7 } { 2 } \left( \tan ^ { - 1 } ( x + 1 ) + \frac { x + 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$

Question 5

Find the integral. Select the correct answer.
$$\int \cos ^ { 5 } x \sin ^ { 2 } x d x$$

Accepted Answer

A)  $\frac { 1 } { 5 } \sin ^ { 5 } x - \frac { 2 } { 3 } \sin ^ { 3 } x + \sin ^ { 3 } x + C$ 
B)  $\frac { 1 } { 7 } \sin ^ { 7 } x - \frac { 2 } { 5 } \sin ^ { 5 } x + \frac { 1 } { 3 } \sin ^ { 3 } x + C$ 
C)  $\frac { 1 } { 5 } \sin ^ { 5 } x + \frac { 2 } { 3 } \sin ^ { 3 } x - \sin ^ { 3 } x + C$ 
D)  $\frac { 1 } { 7 } \sin ^ { 2 } x + \frac { 2 } { 5 } \sin ^ { 5 } x - \frac { 1 } { 3 } \sin ^ { 3 } x + C$ 
A)  $\frac { 1 } { 5 } \sin ^ { 5 } x - \frac { 2 } { 3 } \sin ^ { 3 } x + \sin ^ { 3 } x + C$ 
B)  $\frac { 1 } { 7 } \sin ^ { 7 } x - \frac { 2 } { 5 } \sin ^ { 5 } x + \frac { 1 } { 3 } \sin ^ { 3 } x + C$ 
C)  $\frac { 1 } { 5 } \sin ^ { 5 } x + \frac { 2 } { 3 } \sin ^ { 3 } x - \sin ^ { 3 } x + C$ 
D)  $\frac { 1 } { 7 } \sin ^ { 2 } x + \frac { 2 } { 5 } \sin ^ { 5 } x - \frac { 1 } { 3 } \sin ^ { 3 } x + C$

Question 6

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

Accepted Answer

The answer of Find the integral.
\[\int \frac { 3 x...

Question 7

Evaluate the integral.
$$\int _ { 4 } ^ { \sqrt { 17 } } \sqrt { x ^ { 2 } - 16 } d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { 4 }...

Question 8

Evaluate the integral using an appropriate trigonometric substitution.
$$\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$$

Accepted Answer

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

Question 9

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 10

Let $a$ and $b$ be real numbers. What integral must appear in place of the question mark "?" to make the following statement true?
$$\int _ { - \infty } ^ { a } \frac { 10 } { x ^ { 2 } + 9 } d x + \int _ { a } ^ { \infty } \frac { 10 } { x ^ { 2 } + 9 } d x = ? + \int _ { \partial } ^ { \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$$

Accepted Answer

A)  $\int _ { - \infty } ^ { a } \frac { 9 } { x ^ { 2 } + 10 } d x$ 
B)  $\int _ { 0 } ^ { - \infty } \frac { 10 } { x ^ { 2 } - 9 } d x$ 
C)  $\int _ { - \infty } ^ { b } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
D)  $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
E)  none of these 
A)  $\int _ { - \infty } ^ { a } \frac { 9 } { x ^ { 2 } + 10 } d x$ 
B)  $\int _ { 0 } ^ { - \infty } \frac { 10 } { x ^ { 2 } - 9 } d x$ 
C)  $\int _ { - \infty } ^ { b } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
D)  $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
E)  none of these

Question 11

Evaluate the integral.
$$\int _ { 0 } ^ { 1 } \frac { 3 } { x ^ { 03 } } d x$$

Accepted Answer

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

Question 12

Estimate the area of the shaded region by using the Trapezoidal Rule with $n = 2$. Round the answer to the nearest tenth.

Accepted Answer

The answer of Estimate the area of the shaded region...

Question 13

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 14

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$-axis. Find the volume of the resulting solid.

Accepted Answer

The answer of The region under the curve \(y =...

Question 15

Use the Table of Integrals to evaluate the integral.
$$\int e ^ { 8 x } \sin 4 x d x$$

Accepted Answer

The answer of Use the Table of Integrals to evaluate...

Question 16

Evaluate the integral.
$$\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$$

Accepted Answer

A)  $30 e ^ { - 4 } + 10$ 
B)  $- 30 e ^ { 4 } + 10$ 
C)  $- 90 e ^ { - 8 } + 10$ 
D)  $- 30 e ^ { - 4 } - 1$ 
E)  None of these 
A)  $30 e ^ { - 4 } + 10$ 
B)  $- 30 e ^ { 4 } + 10$ 
C)  $- 90 e ^ { - 8 } + 10$ 
D)  $- 30 e ^ { - 4 } - 1$ 
E)  None of these

Question 17

Evaluate the integral. Select the correct answer.
$$\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$$

Accepted Answer

A)  $30 e ^ { - 4 } + 10$ 
B)  $- 30 e ^ { 4 } + 10$ 
C)  $- 90 e ^ { - 8 } + 10$ 
D)  $- 30 e ^ { - 4 } - 1$ 
E)  None of these 
A)  $30 e ^ { - 4 } + 10$ 
B)  $- 30 e ^ { 4 } + 10$ 
C)  $- 90 e ^ { - 8 } + 10$ 
D)  $- 30 e ^ { - 4 } - 1$ 
E)  None of these

Question 18

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places.
$$\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$$

Accepted Answer

The answer of Use Simpson's Rule to approximate the integral...

Question 19

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 _ { 0...

Question 20

Evaluate the integral.
$\int _ { 0 } ^ { 4 } \frac { x } { x + 8 } d x$

Accepted Answer

A)  $12 - 8 \ln 12 + \ln \sqrt { 8 }$ 
B)  $\frac { 1 } { 8 \ln 12 } - \ln \sqrt { 8 }$ 
C)  $4 - 12 \ln 8 + \sqrt { 8 } \ln 16$ 
D)  $4 - 8 ( \ln 12 - \ln 8 )$ 
E)  None of these 
A)  $12 - 8 \ln 12 + \ln \sqrt { 8 }$ 
B)  $\frac { 1 } { 8 \ln 12 } - \ln \sqrt { 8 }$ 
C)  $4 - 12 \ln 8 + \sqrt { 8 } \ln 16$ 
D)  $4 - 8 ( \ln 12 - \ln 8 )$ 
E)  None of these

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$

Use the Midpoint Rule to approximate the given integral with the specified value of $n$ . Compare your result to the actual value. Find the error in the approximation. $2 \int _ { 2 } ^ { 3 } e ^ { - \sqrt { x } } d x , n = 6$

Evaluate the integral. $\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }$

Find the integral. Select the correct answer. $\int \cos ^ { 5 } x \sin ^ { 2 } x d x$

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

Evaluate the integral. $\int _ { 4 } ^ { \sqrt { 17 } } \sqrt { x ^ { 2 } - 16 } d x$

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$

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

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { 3 } { x ^ { 03 } } d x$

Estimate the area of the shaded region by using the Trapezoidal Rule with $n = 2$ . Round the answer to the nearest tenth.

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.

The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$ -axis. Find the volume of the resulting solid.

Use the Table of Integrals to evaluate the integral. $\int e ^ { 8 x } \sin 4 x d x$

Evaluate the integral. $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$

Evaluate the integral. Select the correct answer. $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$

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

Evaluate the integral. $\int _ { 0 } ^ { 4 } \frac { x } { x + 8 } d x$

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Exam 7: Techniques of Integration

Find the average value of the function f(x)f ( x )f(x) in the interval [−π,π][ - \pi , \pi ][−π,π] . f(x)=sin⁡6xcos⁡5xf ( x ) = \sin ^ { 6 } x \cos ^ { 5 } xf(x)=sin6xcos5x

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. ∫01dx2x+4;n=7\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7∫01​2x+4dx​;n=7

Use the Midpoint Rule to approximate the given integral with the specified value of nnn . Compare your result to the actual value. Find the error in the approximation. 2∫23e−xdx,n=62 \int _ { 2 } ^ { 3 } e ^ { - \sqrt { x } } d x , n = 62∫23​e−x​dx,n=6

Evaluate the integral. ∫7dx(x2+2x+2)2\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }∫(x2+2x+2)27dx​

Find the integral. Select the correct answer. ∫cos⁡5xsin⁡2xdx\int \cos ^ { 5 } x \sin ^ { 2 } x d x∫cos5xsin2xdx

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

Evaluate the integral. ∫417x2−16dx\int _ { 4 } ^ { \sqrt { 17 } } \sqrt { x ^ { 2 } - 16 } d x∫417​​x2−16​dx

Evaluate the integral using an appropriate trigonometric substitution. ∫13x2−1x4dx\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x∫13​x4x2−1​​dx

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. ∫013x03dx\int _ { 0 } ^ { 1 } \frac { 3 } { x ^ { 03 } } d x∫01​x033​dx

Estimate the area of the shaded region by using the Trapezoidal Rule with n=2n = 2n=2 . Round the answer to the nearest tenth.

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.

The region under the curve y=3sin⁡2x,0≤x≤πy = 3 \sin ^ { 2 } x , 0 \leq x \leq \piy=3sin2x,0≤x≤π is rotated about the xxx -axis. Find the volume of the resulting solid.

Use the Table of Integrals to evaluate the integral. ∫e8xsin⁡4xdx\int e ^ { 8 x } \sin 4 x d x∫e8xsin4xdx

Evaluate the integral. ∫08(x2+8)e−xdx\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x∫08​(x2+8)e−xdx

Evaluate the integral. Select the correct answer. ∫08(x2+8)e−xdx\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x∫08​(x2+8)e−xdx

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. ∫−11x2+1dx;n=6\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6∫−11​x2+1​dx;n=6

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

Evaluate the integral. ∫04xx+8dx\int _ { 0 } ^ { 4 } \frac { x } { x + 8 } d x∫04​x+8x​dx