Question 1

Integrate the function.
-$$\int \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x ^ { 2 } } d x$$

Accepted Answer

A)  $x + \ln \left| \sqrt { x ^ { 2 } + 36 } \right| + \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x } + C$ 
B)  $\frac { 1 } { 9 } \ln \left| \sqrt { x ^ { 2 } + 36 } + x \right| - \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x } + C$ 
C)  $\frac { 1 } { 9 } \ln \left| \sqrt { x ^ { 2 } + 36 } + x \right| - \sin ^ { - 1 } \frac { x } { 6 } + C$ 
D)  $\ln \left| \sqrt { x ^ { 2 } + 36 } + x \right| + \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x } + C$ 
A)  $x + \ln \left| \sqrt { x ^ { 2 } + 36 } \right| + \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x } + C$ 
B)  $\frac { 1 } { 9 } \ln \left| \sqrt { x ^ { 2 } + 36 } + x \right| - \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x } + C$ 
C)  $\frac { 1 } { 9 } \ln \left| \sqrt { x ^ { 2 } + 36 } + x \right| - \sin ^ { - 1 } \frac { x } { 6 } + C$ 
D)  $\ln \left| \sqrt { x ^ { 2 } + 36 } + x \right| + \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x } + C$ B

Question 2

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 4 x ^ { 4 } + 36 x ^ { 2 } + 72 } { x \left( x ^ { 2 } + 6 \right) ^ { 2 } } d x$$

Accepted Answer

A)  $2 \ln | x | + \ln \left| x ^ { 2 } + 6 \right| - \frac { 4 } { x ^ { 2 } + 6 } + C$ 
B)  $2 \ln | x | - \frac { 4 } { x ^ { 2 } + 6 } + C$ 
C)  $2 \ln | x | + \ln \left| x ^ { 2 } + 6 \right| + C$ 
D)  $8 \ln | x | + \ln \left| x ^ { 2 } + 6 \right| + C$ 
A)  $2 \ln | x | + \ln \left| x ^ { 2 } + 6 \right| - \frac { 4 } { x ^ { 2 } + 6 } + C$ 
B)  $2 \ln | x | - \frac { 4 } { x ^ { 2 } + 6 } + C$ 
C)  $2 \ln | x | + \ln \left| x ^ { 2 } + 6 \right| + C$ 
D)  $8 \ln | x | + \ln \left| x ^ { 2 } + 6 \right| + C$ C

Question 3

Find the area or volume.
-$\text { Find the area of the region bounded by the curve } y = 8 x ^ { - 2 } \text {, the } x \text {-axis, and on the left by } x = 1 \text {. }$

Accepted Answer

A)  64 
B)  16 
C)  8 
D)  4 
A)  64 
B)  16 
C)  8 
D)  4 C

Question 4

Use a CAS to perform the integration.
-$\int \frac { \ln x } { x ^ { 8 } }$

Accepted Answer

A)  $\frac { - 7 \ln x - 1 } { 7 x ^ { 7 } }$ 
B)  $\frac { - 7 \ln x - 1 } { 49 x ^ { 7 } }$ 
C)  $\frac { - \ln x - 1 } { 49 x ^ { 7 } }$ 
D)  $\frac { - 6 \ln x - 1 } { 36 x ^ { 6 } }$ 
A)  $\frac { - 7 \ln x - 1 } { 7 x ^ { 7 } }$ 
B)  $\frac { - 7 \ln x - 1 } { 49 x ^ { 7 } }$ 
C)  $\frac { - \ln x - 1 } { 49 x ^ { 7 } }$ 
D)  $\frac { - 6 \ln x - 1 } { 36 x ^ { 6 } }$

Question 5

Use reduction formulas to evaluate the integral.
-$$\int \cot ^ { 4 } 3 x d x$$

Accepted Answer

A)  $- \frac { 1 } { 3 } \cot ^ { 3 } 3 x - \frac { 1 } { 3 } \cot 3 x + x + C$ 
B)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + x + C$ 
C)  $- \frac { 1 } { 9 } \cot ^ { 2 } 3 x + \frac { 1 } { 3 } \cot 3 x - x + C$ 
D)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + C$ 
A)  $- \frac { 1 } { 3 } \cot ^ { 3 } 3 x - \frac { 1 } { 3 } \cot 3 x + x + C$ 
B)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + x + C$ 
C)  $- \frac { 1 } { 9 } \cot ^ { 2 } 3 x + \frac { 1 } { 3 } \cot 3 x - x + C$ 
D)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + C$

Question 6

Find the surface area or volume.
-Use substitution and a table of integrals to find, to two decimal places, the area of the surface generated by revolving the curve $y = e ^ { x } , 0 \leq x \leq 3$, about the $x$-axis.

Accepted Answer

A)  1666.81 
B)  1544.52 
C)  1381.47 
D)  1276.14 
A)  1666.81 
B)  1544.52 
C)  1381.47 
D)  1276.14

Question 7

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 0 } ^ { \infty } 21 x e ^ { 2 x } d x$$

Accepted Answer

A)  1.3333 
B)  Divergent 
C)  1.6667 
D)  2.6667 
A)  1.3333 
B)  Divergent 
C)  1.6667 
D)  2.6667

Question 8

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions.
-$$\int \frac { x ^ { 4 } } { x ^ { 2 } - 9 } d x$$

Accepted Answer

A)  $\frac { x ^ { 3 } } { 3 } + 9 x + \frac { 27 } { 2 } \ln | x - 9 | - \frac { 27 } { 2 } \ln | x + 9 | + C$ 
B)  $\frac { x ^ { 3 } } { 3 } + 9 x + \frac { 27 } { 2 } \ln | x - 3 | - \frac { 27 } { 2 } \ln | x + 3 | + C$ 
C)  $\frac { x ^ { 3 } } { 3 } + \frac { 9 } { 2 } \ln | x - 3 | - \frac { 9 } { 2 } \ln | x + 3 | + C$ 
D)  $\frac { x ^ { 3 } } { 3 } + 9 x - \frac { 27 } { 2 } \ln | x - 3 | + \frac { 27 } { 2 } \ln | x + 3 | + C$ 
A)  $\frac { x ^ { 3 } } { 3 } + 9 x + \frac { 27 } { 2 } \ln | x - 9 | - \frac { 27 } { 2 } \ln | x + 9 | + C$ 
B)  $\frac { x ^ { 3 } } { 3 } + 9 x + \frac { 27 } { 2 } \ln | x - 3 | - \frac { 27 } { 2 } \ln | x + 3 | + C$ 
C)  $\frac { x ^ { 3 } } { 3 } + \frac { 9 } { 2 } \ln | x - 3 | - \frac { 9 } { 2 } \ln | x + 3 | + C$ 
D)  $\frac { x ^ { 3 } } { 3 } + 9 x - \frac { 27 } { 2 } \ln | x - 3 | + \frac { 27 } { 2 } \ln | x + 3 | + C$

Question 9

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 6 } ^ { \infty } \frac { d x } { ( x - 5 ) ( x - 4 ) }$$

Accepted Answer

A)  $\ln 5$ 
B)  $\frac { 1 } { 2 } \ln 6$ 
C)  $\ln 2$ 
D)  $- \frac { 1 } { 5 } \ln 2$ 
A)  $\ln 5$ 
B)  $\frac { 1 } { 2 } \ln 6$ 
C)  $\ln 2$ 
D)  $- \frac { 1 } { 5 } \ln 2$

Question 10

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 ounces and a standard
Deviation of 0.04 ounce. Find the probability that the bottle contains fewer than 12.04 ounces of beer.

Accepted Answer

A)  0.4938 
B)  0.9938 
C)  0.0062 
D)  0.5062 
A)  0.4938 
B)  0.9938 
C)  0.0062 
D)  0.5062

Question 11

Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 0 } ^ { \pi } 4 \cos x d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Accepted Answer

A)  19 
B)  18 
C)  20 
D)  16 
A)  19 
B)  18 
C)  20 
D)  16

Question 12

Provide an appropriate response.
-(a) Show that $\int _ { 0 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$ converges.
(b) Show that $\int _ { 50 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 } \leq 0.0002$.
(c) Suppose $\int _ { 0 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$ is approximated by $\int _ { 0 } ^ { 50 } \frac { d x } { x ^ { 3 } + 1 }$. Based on your answer to part (b), what is the maximum possible error?
(d) Use a numerical method to estimate the value of $\int _ { 0 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$.
(e) Determine whether $\int _ { - 1 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$ converges or diverges, and justify your answer. If it converges, estimate its value to an accuracy of at least two decimal places.

Accepted Answer

(a) Possible answer:
Note that

Question 13

Evaluate the integral.
-$$\int 2 \cos ^ { 3 } 3 x d x$$

Accepted Answer

A)  $\frac { 2 } { 3 } \sin 3 x + \frac { 2 } { 9 } \sin ^ { 3 } 3 x + C$ 
B)  $\frac { 2 } { 3 } \sin 3 x - \frac { 2 } { 9 } \cos ^ { 3 } 3 x + C$ 
C)  $\frac { 2 } { 3 } \sin 3 x - \frac { 2 } { 9 } \sin ^ { 3 } 3 x + C$ 
D)  $2 \sin 3 x - \frac { 2 } { 3 } \sin ^ { 3 } 3 x + C$ 
A)  $\frac { 2 } { 3 } \sin 3 x + \frac { 2 } { 9 } \sin ^ { 3 } 3 x + C$ 
B)  $\frac { 2 } { 3 } \sin 3 x - \frac { 2 } { 9 } \cos ^ { 3 } 3 x + C$ 
C)  $\frac { 2 } { 3 } \sin 3 x - \frac { 2 } { 9 } \sin ^ { 3 } 3 x + C$ 
D)  $2 \sin 3 x - \frac { 2 } { 3 } \sin ^ { 3 } 3 x + C$

Question 14

Solve the initial value problem for x as a function of t.
-$$\left( t ^ { 2 } - 5 t + 6 \right) \frac { d x } { d t } = 1 \quad ( t > 3 ) , x ( 4 ) = 0$$

Accepted Answer

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

Question 15

Solve the problem.
-Find the area of the region enclosed by the curve $y = x \cos x$ and the $x$-axis for $\frac { 9 } { 2 } \pi \leq x \leq \frac { 11 } { 2 } \pi$.

Accepted Answer

A)  $9 \pi$ 
B)  $11 \pi$ 
C)  9 
D)  $10 \pi$ 
A)  $9 \pi$ 
B)  $11 \pi$ 
C)  9 
D)  $10 \pi$

Question 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.
-$f ( x ) = 6 e ^ { - 5 x }$ over $[ 0 , k ]$

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

Use Simpson's Rule with n = 4 steps to estimate the integral.
-$$\int _ { 0 } ^ { 2 } 5 x ^ { 2 } d x$$

Accepted Answer

A)  $\frac { 40 } { 3 }$ 
B)  $\frac { 55 } { 4 }$ 
C)  $\frac { 35 } { 3 }$ 
D)  $\frac { 20 } { 3 }$ 
A)  $\frac { 40 } { 3 }$ 
B)  $\frac { 55 } { 4 }$ 
C)  $\frac { 35 } { 3 }$ 
D)  $\frac { 20 } { 3 }$

Question 19

Use integration by parts to establish a reduction formula for the integral.
-$\int ( \ln a x ) ^ { n } d x$

Accepted Answer

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

Question 20

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

Accepted Answer

A)  $\frac { \sec ^ { 6 } 2 x } { 6 } + C$ 
B)  $\frac { \tan ^ { 4 } 2 x \sec 2 x } { 8 } + C$ 
C)  $\frac { \tan ^ { 4 } 2 x } { 8 } + C$ 
D)  $\frac { \tan ^ { 6 } 2 x } { 6 } + C$ 
A)  $\frac { \sec ^ { 6 } 2 x } { 6 } + C$ 
B)  $\frac { \tan ^ { 4 } 2 x \sec 2 x } { 8 } + C$ 
C)  $\frac { \tan ^ { 4 } 2 x } { 8 } + C$ 
D)  $\frac { \tan ^ { 6 } 2 x } { 6 } + C$

Integrate the function. - $\int \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x ^ { 2 } } d x$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 4 x ^ { 4 } + 36 x ^ { 2 } + 72 } { x \left( x ^ { 2 } + 6 \right) ^ { 2 } } d x$

Find the area or volume. - $\text { Find the area of the region bounded by the curve } y = 8 x ^ { - 2 } \text {, the } x \text {-axis, and on the left by } x = 1 \text {. }$

Use a CAS to perform the integration. - $\int \frac { \ln x } { x ^ { 8 } }$

Use reduction formulas to evaluate the integral. - $\int \cot ^ { 4 } 3 x d x$

Find the surface area or volume. -Use substitution and a table of integrals to find, to two decimal places, the area of the surface generated by revolving the curve $y = e ^ { x } , 0 \leq x \leq 3$ , about the $x$ -axis.

Evaluate the improper integral or state that it is divergent. - $\int _ { 0 } ^ { \infty } 21 x e ^ { 2 x } d x$

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - $\int \frac { x ^ { 4 } } { x ^ { 2 } - 9 } d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { 6 } ^ { \infty } \frac { d x } { ( x - 5 ) ( x - 4 ) }$

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 0 } ^ { \pi } 4 \cos x d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Evaluate the integral. - $\int 2 \cos ^ { 3 } 3 x d x$

Solve the initial value problem for x as a function of t. - $\left( t ^ { 2 } - 5 t + 6 \right) \frac { d x } { d t } = 1 \quad ( t > 3 ) , x ( 4 ) = 0$

Solve the problem. -Find the area of the region enclosed by the curve $y = x \cos x$ and the $x$ -axis for $\frac { 9 } { 2 } \pi \leq x \leq \frac { 11 } { 2 } \pi$ .

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. - $f ( x ) = 6 e ^ { - 5 x }$ over $[ 0 , k ]$

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

Use Simpson's Rule with n = 4 steps to estimate the integral. - $\int _ { 0 } ^ { 2 } 5 x ^ { 2 } d x$

Use integration by parts to establish a reduction formula for the integral. - $\int ( \ln a x ) ^ { n } d x$

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

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Exam 9: Techniques of Integration

Integrate the function. - ∫x2+369x2dx\int \frac { \sqrt { x ^ { 2 } + 36 } } { 9 x ^ { 2 } } d x∫9x2x2+36​​dx

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫4x4+36x2+72x(x2+6)2dx\int \frac { 4 x ^ { 4 } + 36 x ^ { 2 } + 72 } { x \left( x ^ { 2 } + 6 \right) ^ { 2 } } d x∫x(x2+6)24x4+36x2+72​dx

Use a CAS to perform the integration. - ∫ln⁡xx8\int \frac { \ln x } { x ^ { 8 } }∫x8lnx​

Use reduction formulas to evaluate the integral. - ∫cot⁡43xdx\int \cot ^ { 4 } 3 x d x∫cot43xdx

Find the surface area or volume. -Use substitution and a table of integrals to find, to two decimal places, the area of the surface generated by revolving the curve y=ex,0≤x≤3y = e ^ { x } , 0 \leq x \leq 3y=ex,0≤x≤3 , about the xxx -axis.

Evaluate the improper integral or state that it is divergent. - ∫0∞21xe2xdx\int _ { 0 } ^ { \infty } 21 x e ^ { 2 x } d x∫0∞​21xe2xdx

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - ∫x4x2−9dx\int \frac { x ^ { 4 } } { x ^ { 2 } - 9 } d x∫x2−9x4​dx

Evaluate the improper integral or state that it is divergent. - ∫6∞dx(x−5)(x−4)\int _ { 6 } ^ { \infty } \frac { d x } { ( x - 5 ) ( x - 4 ) }∫6∞​(x−5)(x−4)dx​

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral ∫0π4cos⁡xdx\int _ { 0 } ^ { \pi } 4 \cos x d x∫0π​4cosxdx with an error of magnitude less than 10−410 ^ { - 4 }10−4 using Simpson's Rule.

Evaluate the integral. - ∫2cos⁡33xdx\int 2 \cos ^ { 3 } 3 x d x∫2cos33xdx

Solve the initial value problem for x as a function of t. - (t2−5t+6)dxdt=1(t>3),x(4)=0\left( t ^ { 2 } - 5 t + 6 \right) \frac { d x } { d t } = 1 \quad ( t > 3 ) , x ( 4 ) = 0(t2−5t+6)dtdx​=1(t>3),x(4)=0

Solve the problem. -Find the area of the region enclosed by the curve y=xcos⁡xy = x \cos xy=xcosx and the xxx -axis for 92π≤x≤112π\frac { 9 } { 2 } \pi \leq x \leq \frac { 11 } { 2 } \pi29​π≤x≤211​π .

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)=6e−5xf ( x ) = 6 e ^ { - 5 x }f(x)=6e−5x over [0,k][ 0 , k ][0,k]

Use integration by parts to establish a reduction formula for the integral. - ∫sin⁡nxdx\int \sin ^ { n } x d x∫sinnxdx

Use Simpson's Rule with n = 4 steps to estimate the integral. - ∫025x2dx\int _ { 0 } ^ { 2 } 5 x ^ { 2 } d x∫02​5x2dx

Use integration by parts to establish a reduction formula for the integral. - ∫(ln⁡ax)ndx\int ( \ln a x ) ^ { n } d x∫(lnax)ndx

Use reduction formulas to evaluate the integral. - ∫sin⁡32xsec⁡52xdx\int \sin ^ { 3 } 2 x \sec ^ { 5 } 2 x d x∫sin32xsec52xdx