Question 1

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

Accepted Answer

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

Question 2

Evaluate the integral.
-$$\int _ { 1 } ^ { \mathrm { e } ^ { 3 } } \frac { \ln ^ { 3 } \left( \mathrm { x } ^ { 3 } \right) } { \mathrm { x } } \mathrm { dx }$$

Accepted Answer

A)  $3 \mathrm { e } ^ { 9 }$ 
B)  $\frac { 2187 } { 4 } \mathrm { e } ^ { 3 }$ 
C)  $\frac { 2187 } { 4 }$ 
D)  $\frac { 19683 } { 4 }$ 
A)  $3 \mathrm { e } ^ { 9 }$ 
B)  $\frac { 2187 } { 4 } \mathrm { e } ^ { 3 }$ 
C)  $\frac { 2187 } { 4 }$ 
D)  $\frac { 19683 } { 4 }$

Question 3

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

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  2 
C)  $\sqrt { 2 }$ 
D)  $2 \sqrt { 2 }$ 
A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  2 
C)  $\sqrt { 2 }$ 
D)  $2 \sqrt { 2 }$

Question 4

Provide an appropriate response.
-(a) Show that $\int _ { 2 } ^ { \infty } e ^ { - 2 x } \mathrm { dx } = \frac { 1 } { 2 } \mathrm { e } ^ { - 4 } < 0.0092$ and hence that $\int _ { 2 } ^ { \infty } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx } < 0.0092$.
(b) Explain why this means that $\int _ { 0 } ^ { \infty } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx }$ can be replaced by $\int _ { 0 } ^ { 2 } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx }$ without introducing an error of magnitude greater than $0.0092 .$

Accepted Answer

(a) @#LAT-DLM&

(b) Since @#LAT-DLM& for

Question 5

Solve the problem.
-Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 0 } ^ { \pi } 4 \mathrm { x } \sin \mathrm { x } \mathrm {} \mathrm { dx }$ with $\mathrm { n } = 6$ steps. Give your answer as a decimal rounded to four decimal places.

Accepted Answer

A)  0.2871 
B)  0.1435 
C)  0.9019 
D)  0.5742 
A)  0.2871 
B)  0.1435 
C)  0.9019 
D)  0.5742

Question 6

Evaluate the integral by making a substitution and then using a table of integrals.
-$$\int \frac { x d x } { 25 x ^ { 2 } + 20 x + 4 }$$

Accepted Answer

A)  $\frac { 1 } { 5 } \ln \left( 5 x ^ { 2 } + 2 x \right)$ 
B)  $\frac { 1 } { 4 } \left[ \ln ( 2 x + 5 ) + \frac { 5 } { 2 x + 5 } \right] + C$ 
C)  $\frac { x } { 5 } - \frac { 2 } { 25 } \ln ( 5 x + 2 ) + C$ 
D)  $\frac { 1 } { 25 } \left[ \ln ( 5 x + 2 ) + \frac { 2 } { 5 x + 2 } \right] + C$ 
A)  $\frac { 1 } { 5 } \ln \left( 5 x ^ { 2 } + 2 x \right)$ 
B)  $\frac { 1 } { 4 } \left[ \ln ( 2 x + 5 ) + \frac { 5 } { 2 x + 5 } \right] + C$ 
C)  $\frac { x } { 5 } - \frac { 2 } { 25 } \ln ( 5 x + 2 ) + C$ 
D)  $\frac { 1 } { 25 } \left[ \ln ( 5 x + 2 ) + \frac { 2 } { 5 x + 2 } \right] + C$

Question 7

Solve the problem by integration.
-By a computer analysis, the electric current i (in A) in a certain circuit is given by i $= \frac { 0.0040 \left( 8 t ^ { 2 } + 15 t + 75 \right) } { ( t + 5 ) \left( t ^ { 2 } + 15 \right) }$, where $t$ is the time (in s). Find the total charge that passes a point in the circuit in the first $0.25 \mathrm {~s}$.

Accepted Answer

A)  0.0002 C 
B)  0.0051 C 
C)  0.0010 C 
D)  0.0049 C 
A)  0.0002 C 
B)  0.0051 C 
C)  0.0010 C 
D)  0.0049 C

Question 8

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { x + 9 } { x ^ { 2 } + 6 x } d x$$

Accepted Answer

A)  $\frac { 3 } { 2 } \ln \left| x ^ { 9 } ( x + 6 ) ^ { 3 } \right| + C$ 
B)  $\ln \left| \frac { x ^ { 9 } } { ( x + 6 ) ^ { 3 } } \right| + C$ 
C)  $\frac { 1 } { 6 } \ln \left| x ^ { 9 } ( x + 6 ) ^ { 3 } \right| + C$ 
D)  $\frac { 1 } { 6 } \ln \left| \frac { x ^ { 9 } } { ( x + 6 ) ^ { 3 } } \right| + C$ 
A)  $\frac { 3 } { 2 } \ln \left| x ^ { 9 } ( x + 6 ) ^ { 3 } \right| + C$ 
B)  $\ln \left| \frac { x ^ { 9 } } { ( x + 6 ) ^ { 3 } } \right| + C$ 
C)  $\frac { 1 } { 6 } \ln \left| x ^ { 9 } ( x + 6 ) ^ { 3 } \right| + C$ 
D)  $\frac { 1 } { 6 } \ln \left| \frac { x ^ { 9 } } { ( x + 6 ) ^ { 3 } } \right| + C$

Question 9

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

Accepted Answer

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

Question 10

Determine whether the improper integral converges or diverges.
-$$\int _ { 1 } ^ { \infty } \frac { 8 \mathrm { e } ^ { x } } { \ln x } d x$$

Accepted Answer

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

Question 11

Find the area or volume.
-Find the volume of the solid generated by revolving the area under $y = 6 e ^ { - x }$ in the first quadrant about the y-axis.

Accepted Answer

A)  $12 \pi$ 
B)  $6 \pi$ 
C)  $24 \pi$ 
D)  1 
A)  $12 \pi$ 
B)  $6 \pi$ 
C)  $24 \pi$ 
D)  1

Question 12

Solve the problem.
-Find the length of the curve $y = \ln ( \csc x ) , \pi / 3 \leq x \leq \pi / 2$

Accepted Answer

A)  $\ln ( \sqrt { 3 } )$ 
B)  $\ln ( \sqrt { 3 } + 1 )$ 
C)  $1 - \ln ( \sqrt { 3 } )$ 
D)  $\ln ( 2 \sqrt { 3 } )$ 
A)  $\ln ( \sqrt { 3 } )$ 
B)  $\ln ( \sqrt { 3 } + 1 )$ 
C)  $1 - \ln ( \sqrt { 3 } )$ 
D)  $\ln ( 2 \sqrt { 3 } )$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Determine whether the improper integral converges or diverges.
-$$\int _ { - 9 } ^ { 10 } \frac { d x } { ( x + 1 ) ^ { 1 / 3 } }$$

Accepted Answer

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

Question 16

Evaluate the integral.
-$$\int \frac { \cos t d t } { \sin ^ { 2 } t - 4 \sin t + 3 }$$

Accepted Answer

A)  $\frac { 1 } { 2 } \ln | \mathrm { t } - 3 | - \frac { 1 } { 2 } \ln | \mathrm { t } - 1 | + \mathrm { C }$ 
B)  $\frac { 1 } { 2 } \ln | \sin t - 3 | + \frac { 1 } { 2 } \ln | \sin t - 1 | + C$ 
C)  $\frac { 1 } { 2 } \ln | \sin t - 3 | - \frac { 1 } { 2 } \ln | \sin t - 1 | + C$ 
D)  $\ln | \sin t - 3 | - \ln | \sin t - 1 | + C$ 
A)  $\frac { 1 } { 2 } \ln | \mathrm { t } - 3 | - \frac { 1 } { 2 } \ln | \mathrm { t } - 1 | + \mathrm { C }$ 
B)  $\frac { 1 } { 2 } \ln | \sin t - 3 | + \frac { 1 } { 2 } \ln | \sin t - 1 | + C$ 
C)  $\frac { 1 } { 2 } \ln | \sin t - 3 | - \frac { 1 } { 2 } \ln | \sin t - 1 | + C$ 
D)  $\ln | \sin t - 3 | - \ln | \sin t - 1 | + C$

Question 17

Provide an appropriate response.
-A student claims that $\int _ { a } ^ { b } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }$ always exists, as long as a and b are both positive. Refute this by giving an example of a function for which this is not true.

Accepted Answer

The answer of Provide an appropriate response.
-A student claims that...

Question 18

Use a CAS to perform the integration.
-$$\int x ^ { 6 } \ln x d x$$

Accepted Answer

A)  $\frac { x ^ { 7 } } { 7 } \ln x + \frac { x ^ { 7 } } { 49 } + C$ 
B)  $\frac { x ^ { 7 } } { 7 } \ln x - \frac { x ^ { 8 } } { 56 } + C$ 
C)  $\frac { x ^ { 7 } } { 7 } \ln x - \frac { x ^ { 7 } } { 7 } + C$ 
D)  $\frac { x ^ { 7 } } { 7 } \ln x - \frac { x ^ { 7 } } { 49 } + C$ 
A)  $\frac { x ^ { 7 } } { 7 } \ln x + \frac { x ^ { 7 } } { 49 } + C$ 
B)  $\frac { x ^ { 7 } } { 7 } \ln x - \frac { x ^ { 8 } } { 56 } + C$ 
C)  $\frac { x ^ { 7 } } { 7 } \ln x - \frac { x ^ { 7 } } { 7 } + C$ 
D)  $\frac { x ^ { 7 } } { 7 } \ln x - \frac { x ^ { 7 } } { 49 } + C$

Question 19

Solve the initial value problem for y as a function of x.
-$$\left( 36 - x ^ { 2 } \right) \frac { d y } { d x } = 1 , y ( 0 ) = 3$$

Accepted Answer

A)  $y = \frac { 1 } { 6 } \ln | \sec x + \tan x | + 3$ 
B)  $y = \frac { x } { 36 \sqrt { 36 - x ^ { 2 } } } + 3$ 
C)  $y = \frac { 1 } { 12 } \ln \left| \frac { x + 6 } { x - 6 } \right| + 3$ 
D)  $y = \frac { 1 } { 12 } \ln \left| \frac { x + 6 } { x - 6 } \right|$ 
A)  $y = \frac { 1 } { 6 } \ln | \sec x + \tan x | + 3$ 
B)  $y = \frac { x } { 36 \sqrt { 36 - x ^ { 2 } } } + 3$ 
C)  $y = \frac { 1 } { 12 } \ln \left| \frac { x + 6 } { x - 6 } \right| + 3$ 
D)  $y = \frac { 1 } { 12 } \ln \left| \frac { x + 6 } { x - 6 } \right|$

Question 20

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { x ^ { 2 } + 6 x + 9 } d x$$

Accepted Answer

A)  $36 \ln \left( \frac { 4 } { 3 } \right) - \frac { 35 } { 8 }$ 
B)  $9 \ln 4 - \frac { 71 } { 8 }$ 
C)  $27 \ln \left( \frac { 4 } { 3 } \right) - \frac { 31 } { 4 }$ 
D)  $27 \ln 4 - 9 \ln 3 + \frac { 31 } { 4 }$ 
A)  $36 \ln \left( \frac { 4 } { 3 } \right) - \frac { 35 } { 8 }$ 
B)  $9 \ln 4 - \frac { 71 } { 8 }$ 
C)  $27 \ln \left( \frac { 4 } { 3 } \right) - \frac { 31 } { 4 }$ 
D)  $27 \ln 4 - 9 \ln 3 + \frac { 31 } { 4 }$

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

Evaluate the integral. - $\int _ { 1 } ^ { \mathrm { e } ^ { 3 } } \frac { \ln ^ { 3 } \left( \mathrm { x } ^ { 3 } \right) } { \mathrm { x } } \mathrm { dx }$

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

Solve the problem. -Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 0 } ^ { \pi } 4 \mathrm { x } \sin \mathrm { x } \mathrm {} \mathrm { dx }$ with $\mathrm { n } = 6$ steps. Give your answer as a decimal rounded to four decimal places.

Evaluate the integral by making a substitution and then using a table of integrals. - $\int \frac { x d x } { 25 x ^ { 2 } + 20 x + 4 }$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { x + 9 } { x ^ { 2 } + 6 x } d x$

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

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { 8 \mathrm { e } ^ { x } } { \ln x } d x$

Find the area or volume. -Find the volume of the solid generated by revolving the area under $y = 6 e ^ { - x }$ in the first quadrant about the y-axis.

Solve the problem. -Find the length of the curve $y = \ln ( \csc x ) , \pi / 3 \leq x \leq \pi / 2$

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

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

Determine whether the improper integral converges or diverges. - $\int _ { - 9 } ^ { 10 } \frac { d x } { ( x + 1 ) ^ { 1 / 3 } }$

Evaluate the integral. - $\int \frac { \cos t d t } { \sin ^ { 2 } t - 4 \sin t + 3 }$

Provide an appropriate response. -A student claims that $\int _ { a } ^ { b } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }$ always exists, as long as a and b are both positive. Refute this by giving an example of a function for which this is not true.

Use a CAS to perform the integration. - $\int x ^ { 6 } \ln x d x$

Solve the initial value problem for y as a function of x. - $\left( 36 - x ^ { 2 } \right) \frac { d y } { d x } = 1 , y ( 0 ) = 3$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { x ^ { 2 } + 6 x + 9 } d x$

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Exam 9: Techniques of Integration

Evaluate the integral by making a substitution and then using a table of integrals. - ∫exe2x−25dx\int \frac { e ^ { x } } { e ^ { 2 x } - 25 } d x∫e2x−25ex​dx

Evaluate the integral. - ∫1e3ln⁡3(x3)xdx\int _ { 1 } ^ { \mathrm { e } ^ { 3 } } \frac { \ln ^ { 3 } \left( \mathrm { x } ^ { 3 } \right) } { \mathrm { x } } \mathrm { dx }∫1e3​xln3(x3)​dx

Evaluate the integral. - ∫0π1−cos⁡2xdx\int _ { 0 } ^ { \pi } \sqrt { 1 - \cos 2 x } d x∫0π​1−cos2x​dx

Evaluate the integral by making a substitution and then using a table of integrals. - ∫xdx25x2+20x+4\int \frac { x d x } { 25 x ^ { 2 } + 20 x + 4 }∫25x2+20x+4xdx​

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫x+9x2+6xdx\int \frac { x + 9 } { x ^ { 2 } + 6 x } d x∫x2+6xx+9​dx

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

Determine whether the improper integral converges or diverges. - ∫1∞8exln⁡xdx\int _ { 1 } ^ { \infty } \frac { 8 \mathrm { e } ^ { x } } { \ln x } d x∫1∞​lnx8ex​dx

Find the area or volume. -Find the volume of the solid generated by revolving the area under y=6e−xy = 6 e ^ { - x }y=6e−x in the first quadrant about the y-axis.

Solve the problem. -Find the length of the curve y=ln⁡(csc⁡x),π/3≤x≤π/2y = \ln ( \csc x ) , \pi / 3 \leq x \leq \pi / 2y=ln(cscx),π/3≤x≤π/2

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

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

Determine whether the improper integral converges or diverges. - ∫−910dx(x+1)1/3\int _ { - 9 } ^ { 10 } \frac { d x } { ( x + 1 ) ^ { 1 / 3 } }∫−910​(x+1)1/3dx​

Evaluate the integral. - ∫cos⁡tdtsin⁡2t−4sin⁡t+3\int \frac { \cos t d t } { \sin ^ { 2 } t - 4 \sin t + 3 }∫sin2t−4sint+3costdt​

Provide an appropriate response. -A student claims that ∫abf(x)dx\int _ { a } ^ { b } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }∫ab​f(x)dx always exists, as long as a and b are both positive. Refute this by giving an example of a function for which this is not true.

Use a CAS to perform the integration. - ∫x6ln⁡xdx\int x ^ { 6 } \ln x d x∫x6lnxdx

Solve the initial value problem for y as a function of x. - (36−x2)dydx=1,y(0)=3\left( 36 - x ^ { 2 } \right) \frac { d y } { d x } = 1 , y ( 0 ) = 3(36−x2)dxdy​=1,y(0)=3

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫01x3x2+6x+9dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { x ^ { 2 } + 6 x + 9 } d x∫01​x2+6x+9x3​dx