Question 1

Integrate the function.
-$$\int \frac { y ^ { 2 } } { \left( 16 - y ^ { 2 } \right) ^ { 3 / 2 } } d y$$

Accepted Answer

A)  $\frac { y } { \sqrt { 16 - y ^ { 2 } } } + C$ 
B)  $\frac { y } { \sqrt { 16 - y ^ { 2 } } } - \sin ^ { - 1 } \left( \frac { y } { 4 } \right) + C$ 
C)  $\frac { 4 y } { \sqrt { 16 - y ^ { 2 } } } - \sin ^ { - 1 } y + C$ 
D)  $\sqrt { 16 - y ^ { 2 } } - \sin ^ { - 1 } \left( \frac { y } { 4 } \right) + C$ 
A)  $\frac { y } { \sqrt { 16 - y ^ { 2 } } } + C$ 
B)  $\frac { y } { \sqrt { 16 - y ^ { 2 } } } - \sin ^ { - 1 } \left( \frac { y } { 4 } \right) + C$ 
C)  $\frac { 4 y } { \sqrt { 16 - y ^ { 2 } } } - \sin ^ { - 1 } y + C$ 
D)  $\sqrt { 16 - y ^ { 2 } } - \sin ^ { - 1 } \left( \frac { y } { 4 } \right) + C$

Question 2

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 3 x ^ { 2 } - 11 x + 4 } { x ^ { 3 } - 3 x ^ { 2 } + 2 x } d x$$

Accepted Answer

A)  $2 \ln | x | - 3 \ln | x - 2 | + 4 \ln | x - 1 | + C$ 
B)  $2 \ln | x | + 4 \ln | x - 2 | - 3 \ln | x - 1 | + C$ 
C)  $\ln | x | - \ln | x - 2 | + \ln | x - 1 | + C$ 
D)  $- 3 \ln | x - 2 | + 4 \ln | x - 1 | + C$ 
A)  $2 \ln | x | - 3 \ln | x - 2 | + 4 \ln | x - 1 | + C$ 
B)  $2 \ln | x | + 4 \ln | x - 2 | - 3 \ln | x - 1 | + C$ 
C)  $\ln | x | - \ln | x - 2 | + \ln | x - 1 | + C$ 
D)  $- 3 \ln | x - 2 | + 4 \ln | x - 1 | + C$

Question 3

Use reduction formulas to evaluate the integral.
-$$\int 5 \cot ^ { 3 } x \sin ^ { 2 } x d x$$

Accepted Answer

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

Question 4

For the given probability density function, over the stated interval, find the requested value.
-$f ( x ) = \frac { 1 } { 3 } x ^ { 2 }$, over $[ - 2,4 ] ;$ Find the mean.

Accepted Answer

A)  $\frac { 41 } { 2 }$ 
B)  $\frac { 64 } { 3 }$ 
C)  20 
D)  $\frac { 62 } { 3 }$ 
A)  $\frac { 41 } { 2 }$ 
B)  $\frac { 64 } { 3 }$ 
C)  20 
D)  $\frac { 62 } { 3 }$

Question 5

Use integration by parts to establish a reduction formula for the integral.
-$$\int \cot ^ { n } x d x , n \neq 1$$

Accepted Answer

A)  $\int \cot ^ { n } x d x = \frac { - 1 } { n - 1 } \cot ^ { n - 2 } x - \int \cot ^ { n - 1 } x d x$ 
B)  $\int \cot ^ { n } x d x = \frac { - 1 } { n - 1 } \cot ^ { n - 1 } x - \int \cot ^ { n - 2 } x d x$ 
C)  $\int \cot ^ { n } x d x = - \cot ^ { n - 1 } x + \frac { 1 } { n - 1 } \int \cot ^ { n - 2 } x d x$ 
D)  $\int \cot ^ { n } x d x = \frac { 1 } { n - 1 } \cot ^ { n - 1 } x + \int \cot ^ { n - 1 } x d x$ 
A)  $\int \cot ^ { n } x d x = \frac { - 1 } { n - 1 } \cot ^ { n - 2 } x - \int \cot ^ { n - 1 } x d x$ 
B)  $\int \cot ^ { n } x d x = \frac { - 1 } { n - 1 } \cot ^ { n - 1 } x - \int \cot ^ { n - 2 } x d x$ 
C)  $\int \cot ^ { n } x d x = - \cot ^ { n - 1 } x + \frac { 1 } { n - 1 } \int \cot ^ { n - 2 } x d x$ 
D)  $\int \cot ^ { n } x d x = \frac { 1 } { n - 1 } \cot ^ { n - 1 } x + \int \cot ^ { n - 1 } x d x$

Question 6

For the given probability density function, over the stated interval, find the requested value.
-$f ( x ) = \left\{ \begin{array} { l l } \frac { 7 } { x ^ { 2 } } & 1 \leq x \leq \frac { 7 } { 6 } \ 0 \quad & \text { Otherwise } \end{array} ; \right.$ Find the median.

Accepted Answer

A)  $7 \ln \left( \frac { 7 } { 6 } \right)$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 7 } { 12 }$ 
D)  $\frac { 14 } { 13 }$ 
A)  $7 \ln \left( \frac { 7 } { 6 } \right)$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 7 } { 12 }$ 
D)  $\frac { 14 } { 13 }$

Question 7

Integrate the function.
-$\int \frac { \sqrt { x ^ { 2 } - 9 } } { x } d x$

Accepted Answer

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

Question 8

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 0 } ^ { \infty } 21 e ^ { - 21 x } d x$$

Accepted Answer

A)  Divergent 
B)  1 
C)  0 
D)  -1 
A)  Divergent 
B)  1 
C)  0 
D)  -1

Question 9

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula.
-$$\int _ { 0 } ^ { 1 } 5 \sqrt { x ^ { 2 } + 1 } d x$$

Accepted Answer

A)  $\frac { 5 } { 2 } [ \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) ]$ 
B)  $5 [ \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) ]$ 
C)  $\sqrt { 2 } + \ln ( \sqrt { 2 } + 1 )$ 
D)  $5 [ \sqrt { 2 } - \ln ( \sqrt { 2 } + 1 ) ]$ 
A)  $\frac { 5 } { 2 } [ \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) ]$ 
B)  $5 [ \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) ]$ 
C)  $\sqrt { 2 } + \ln ( \sqrt { 2 } + 1 )$ 
D)  $5 [ \sqrt { 2 } - \ln ( \sqrt { 2 } + 1 ) ]$

Question 10

Evaluate the integral by making a substitution and then using a table of integrals.
-$$\int \frac { d x } { x \left( 36 + ( \ln x ) ^ { 2 } \right) }$$

Accepted Answer

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

Question 11

Solve the problem.
-Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { - 2 x }$, and the line $x = 6$ about the $y$-axis.

Accepted Answer

A)  $\frac { 1 } { 2 } \pi \left( 1 - 13 \mathrm { e } ^ { - 12 } \right)$ 
B)  $- \frac { 1 } { 2 } \pi \left( 1 + 13 \mathrm { e } ^ { - 12 } \right)$ 
C)  $\frac { 1 } { 2 } \pi \left( 1 - 11 \mathrm { e } ^ { - 12 } \right)$ 
D)  $\frac { 1 } { 2 } \pi \left( 1 - 12 \mathrm { e } ^ { - 12 } \right)$ 
A)  $\frac { 1 } { 2 } \pi \left( 1 - 13 \mathrm { e } ^ { - 12 } \right)$ 
B)  $- \frac { 1 } { 2 } \pi \left( 1 + 13 \mathrm { e } ^ { - 12 } \right)$ 
C)  $\frac { 1 } { 2 } \pi \left( 1 - 11 \mathrm { e } ^ { - 12 } \right)$ 
D)  $\frac { 1 } { 2 } \pi \left( 1 - 12 \mathrm { e } ^ { - 12 } \right)$

Question 12

Integrate the function.
-$$\int _ { 0 } ^ { 4 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }$$

Accepted Answer

A)  $65 ^ { 3 / 2 }$ 
B)  $\sqrt { 65 } - 65$ 
C)  $\frac { 4 \sqrt { 65 } } { 65 }$ 
D)  $\frac { \sqrt { 65 } } { 65 }$ 
A)  $65 ^ { 3 / 2 }$ 
B)  $\sqrt { 65 } - 65$ 
C)  $\frac { 4 \sqrt { 65 } } { 65 }$ 
D)  $\frac { \sqrt { 65 } } { 65 }$

Question 13

Determine whether the improper integral converges or diverges.
-$$\int _ { 0 } ^ { 7 } \frac { d x } { 49 - x ^ { 2 } }$$

Accepted Answer

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

Question 14

Determine whether the improper integral converges or diverges.
-$$\int _ { 0 } ^ { 9 } \frac { x ^ { 2 } d x } { \sqrt { 81 - x ^ { 2 } } }$$

Accepted Answer

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

Question 15

Evaluate the integral by making a substitution and then using a table of integrals.
-$$\int e ^ { x } \sqrt { 25 - e ^ { 2 x } } d x$$

Accepted Answer

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

Question 16

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 75 x ^ { 2 } + 50 x + 3 } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { 3 } { 5 } \tan ^ { - 1 } ( 25 x ) + \frac { 1 } { 25 x ^ { 2 } + 1 } + C$ 
B)  $\ln \left| 25 x ^ { 2 } + 1 \right| - \frac { 1 } { 25 x ^ { 2 } + 1 } + C$ 
C)  $\frac { 3 } { 5 } \tan ^ { - 1 } ( 5 x ) - \frac { 1 } { 25 x ^ { 2 } + 1 } + C$ 
D)  $\frac { 3 } { 5 } \tan ^ { - 1 } ( 5 x ) - \frac { 1 } { 25 x ^ { 2 } + 1 } - \frac { 1 } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } } + C$ 
A)  $\frac { 3 } { 5 } \tan ^ { - 1 } ( 25 x ) + \frac { 1 } { 25 x ^ { 2 } + 1 } + C$ 
B)  $\ln \left| 25 x ^ { 2 } + 1 \right| - \frac { 1 } { 25 x ^ { 2 } + 1 } + C$ 
C)  $\frac { 3 } { 5 } \tan ^ { - 1 } ( 5 x ) - \frac { 1 } { 25 x ^ { 2 } + 1 } + C$ 
D)  $\frac { 3 } { 5 } \tan ^ { - 1 } ( 5 x ) - \frac { 1 } { 25 x ^ { 2 } + 1 } - \frac { 1 } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } } + C$

Question 17

Integrate the function.
-$$\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 8 } } d x$$

Accepted Answer

A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } + \tan ^ { - 1 } \left( \frac { x } { 8 } \right) + C$ 
B)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } - 8 \sqrt { x ^ { 2 } + 8 } + C$ 
C)  $\frac { 1 } { 3 } \sqrt { x ^ { 2 } + 8 } - \frac { 8 } { \sqrt { x ^ { 2 } + 8 } } + C$ 
D)  $\frac { 1 } { 8 } \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 8 } + C$ 
A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } + \tan ^ { - 1 } \left( \frac { x } { 8 } \right) + C$ 
B)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } - 8 \sqrt { x ^ { 2 } + 8 } + C$ 
C)  $\frac { 1 } { 3 } \sqrt { x ^ { 2 } + 8 } - \frac { 8 } { \sqrt { x ^ { 2 } + 8 } } + C$ 
D)  $\frac { 1 } { 8 } \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 8 } + C$

Question 18

Provide an appropriate response.
-$\text { The Cauchy density function, } f ( x ) = \frac { 1 } { \pi \left( 1 + x ^ { 2 } \right) } \text {, occurs in probability theory. Show that } \int _ { - \infty } ^ { \infty } \frac { 1 } { \pi \left( 1 + x ^ { 2 } \right) } d x = 1 \text {. }$

Accepted Answer

@#LAT-DLM&. Since @#LAT-DLM& i

Question 19

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 6 } ^ { \infty } \frac { d x } { x ^ { 2 } - 25 }$$

Accepted Answer

A)  $\frac { 1 } { 10 } \ln \frac { 1 } { 6 }$ 
B)  $- \frac { 1 } { 5 } \ln 11$ 
C)  $\frac { 1 } { 10 } \ln 11$ 
D)  $\frac { 1 } { 10 } \ln 6$ 
A)  $\frac { 1 } { 10 } \ln \frac { 1 } { 6 }$ 
B)  $- \frac { 1 } { 5 } \ln 11$ 
C)  $\frac { 1 } { 10 } \ln 11$ 
D)  $\frac { 1 } { 10 } \ln 6$

Question 20

Solve the problem.
-Find the area between $y = ( x - 2 ) e ^ { x }$ and the $x$-axis from $x = 2$ to $x = 5$.

Accepted Answer

A)  $\mathrm { e } ^ { 5 } + \mathrm { e } ^ { 2 }$ 
B)  $2 \mathrm { e } ^ { 5 }$ 
C)  $2 \mathrm { e } ^ { 5 } + \mathrm { e } ^ { 2 }$ 
D)  $\mathrm { e } ^ { 5 } - \mathrm { e } ^ { 2 }$ 
A)  $\mathrm { e } ^ { 5 } + \mathrm { e } ^ { 2 }$ 
B)  $2 \mathrm { e } ^ { 5 }$ 
C)  $2 \mathrm { e } ^ { 5 } + \mathrm { e } ^ { 2 }$ 
D)  $\mathrm { e } ^ { 5 } - \mathrm { e } ^ { 2 }$

Integrate the function. - $\int \frac { y ^ { 2 } } { \left( 16 - y ^ { 2 } \right) ^ { 3 / 2 } } d y$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 3 x ^ { 2 } - 11 x + 4 } { x ^ { 3 } - 3 x ^ { 2 } + 2 x } d x$

Use reduction formulas to evaluate the integral. - $\int 5 \cot ^ { 3 } x \sin ^ { 2 } x d x$

For the given probability density function, over the stated interval, find the requested value. - $f ( x ) = \frac { 1 } { 3 } x ^ { 2 }$ , over $[ - 2,4 ] ;$ Find the mean.

Use integration by parts to establish a reduction formula for the integral. - $\int \cot ^ { n } x d x , n \neq 1$

For the given probability density function, over the stated interval, find the requested value. - $f ( x ) = \left\{ \begin{array} { l l } \frac { 7 } { x ^ { 2 } } & 1 \leq x \leq \frac { 7 } { 6 } \\ 0 \quad & \text { Otherwise } \end{array} ; \right.$ Find the median.

Integrate the function. - $\int \frac { \sqrt { x ^ { 2 } - 9 } } { x } d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { 0 } ^ { \infty } 21 e ^ { - 21 x } d x$

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. - $\int _ { 0 } ^ { 1 } 5 \sqrt { x ^ { 2 } + 1 } d x$

Evaluate the integral by making a substitution and then using a table of integrals. - $\int \frac { d x } { x \left( 36 + ( \ln x ) ^ { 2 } \right) }$

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { - 2 x }$ , and the line $x = 6$ about the $y$ -axis.

Integrate the function. - $\int _ { 0 } ^ { 4 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }$

Determine whether the improper integral converges or diverges. - $\int _ { 0 } ^ { 7 } \frac { d x } { 49 - x ^ { 2 } }$

Determine whether the improper integral converges or diverges. - $\int _ { 0 } ^ { 9 } \frac { x ^ { 2 } d x } { \sqrt { 81 - x ^ { 2 } } }$

Evaluate the integral by making a substitution and then using a table of integrals. - $\int e ^ { x } \sqrt { 25 - e ^ { 2 x } } d x$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 75 x ^ { 2 } + 50 x + 3 } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } } d x$

Integrate the function. - $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 8 } } d x$

Provide an appropriate response. - $\text { The Cauchy density function, } f ( x ) = \frac { 1 } { \pi \left( 1 + x ^ { 2 } \right) } \text {, occurs in probability theory. Show that } \int _ { - \infty } ^ { \infty } \frac { 1 } { \pi \left( 1 + x ^ { 2 } \right) } d x = 1 \text {. }$

Evaluate the improper integral or state that it is divergent. - $\int _ { 6 } ^ { \infty } \frac { d x } { x ^ { 2 } - 25 }$

Solve the problem. -Find the area between $y = ( x - 2 ) e ^ { x }$ and the $x$ -axis from $x = 2$ to $x = 5$ .

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Exam 9: Techniques of Integration

Integrate the function. - ∫y2(16−y2)3/2dy\int \frac { y ^ { 2 } } { \left( 16 - y ^ { 2 } \right) ^ { 3 / 2 } } d y∫(16−y2)3/2y2​dy

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫3x2−11x+4x3−3x2+2xdx\int \frac { 3 x ^ { 2 } - 11 x + 4 } { x ^ { 3 } - 3 x ^ { 2 } + 2 x } d x∫x3−3x2+2x3x2−11x+4​dx

Use reduction formulas to evaluate the integral. - ∫5cot⁡3xsin⁡2xdx\int 5 \cot ^ { 3 } x \sin ^ { 2 } x d x∫5cot3xsin2xdx

For the given probability density function, over the stated interval, find the requested value. - f(x)=13x2f ( x ) = \frac { 1 } { 3 } x ^ { 2 }f(x)=31​x2 , over [−2,4];[ - 2,4 ] ;[−2,4]; Find the mean.

Use integration by parts to establish a reduction formula for the integral. - ∫cot⁡nxdx,n≠1\int \cot ^ { n } x d x , n \neq 1∫cotnxdx,n=1

Integrate the function. - ∫x2−9xdx\int \frac { \sqrt { x ^ { 2 } - 9 } } { x } d x∫xx2−9​​dx

Evaluate the improper integral or state that it is divergent. - ∫0∞21e−21xdx\int _ { 0 } ^ { \infty } 21 e ^ { - 21 x } d x∫0∞​21e−21xdx

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. - ∫015x2+1dx\int _ { 0 } ^ { 1 } 5 \sqrt { x ^ { 2 } + 1 } d x∫01​5x2+1​dx

Evaluate the integral by making a substitution and then using a table of integrals. - ∫dxx(36+(ln⁡x)2)\int \frac { d x } { x \left( 36 + ( \ln x ) ^ { 2 } \right) }∫x(36+(lnx)2)dx​

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y=e−2xy = e ^ { - 2 x }y=e−2x , and the line x=6x = 6x=6 about the yyy -axis.

Integrate the function. - ∫0481dx(81−x2)3/2\int _ { 0 } ^ { 4 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }∫04​(81−x2)3/281dx​

Determine whether the improper integral converges or diverges. - ∫07dx49−x2\int _ { 0 } ^ { 7 } \frac { d x } { 49 - x ^ { 2 } }∫07​49−x2dx​

Determine whether the improper integral converges or diverges. - ∫09x2dx81−x2\int _ { 0 } ^ { 9 } \frac { x ^ { 2 } d x } { \sqrt { 81 - x ^ { 2 } } }∫09​81−x2​x2dx​

Evaluate the integral by making a substitution and then using a table of integrals. - ∫ex25−e2xdx\int e ^ { x } \sqrt { 25 - e ^ { 2 x } } d x∫ex25−e2x​dx

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫75x2+50x+3(25x2+1)2dx\int \frac { 75 x ^ { 2 } + 50 x + 3 } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } } d x∫(25x2+1)275x2+50x+3​dx

Integrate the function. - ∫x3x2+8dx\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 8 } } d x∫x2+8​x3​dx

Evaluate the improper integral or state that it is divergent. - ∫6∞dxx2−25\int _ { 6 } ^ { \infty } \frac { d x } { x ^ { 2 } - 25 }∫6∞​x2−25dx​

Solve the problem. -Find the area between y=(x−2)exy = ( x - 2 ) e ^ { x }y=(x−2)ex and the xxx -axis from x=2x = 2x=2 to x=5x = 5x=5 .