Deck 7: Methods of Integration

Question 1

Integrate $\int 3 \sin ^{2} x \cos x d x$

A) $\frac{3}{2} \sin ^{2} x \cos ^{2} x+C$
B) $\sin ^{2} x \cos ^{2} x+C$
C) $\frac{1}{2} \sin ^{3} \mathrm{x} \cos ^{2} \mathrm{x}+\mathrm{C}$
D) $\sin ^{3} x+C$

Answer 1

$\sin ^{3} x+C$

Question 2

Integrate $\int \frac{4 x}{x^{2}+1} d x$

A) $4 \operatorname{Arctan} x+C$
B) $2 \ln \left(x^{2}+1\right)+C$
C) $4 \ln \left(x^{2}+1\right)+C$
D) $\frac{1}{2} \ln \left(x^{2}+1\right)+C$

Answer 2

$2 \ln \left(x^{2}+1\right)+C$

Question 3

Integrate $\int e^{3 x-1} d x$

A) $\frac{1}{3 x} e^{3 x}+C$
B) $\frac{1}{3} e^{3 x-1}+C$
C) $\mathrm{e}^{3 \mathrm{x}-1}+\mathrm{C}$
D) $e^{3 x}+C$

Answer 3

$\frac{1}{3} e^{3 x-1}+C$

Question 4

Integrate $\int \cos ^{2} 4 x d x$

A) $\frac{1}{2} x+\frac{1}{16} \sin 8 x+C$
B) $\frac{1}{12} \cos ^{3} 4 \mathrm{x}+\mathrm{C}$
C) $\frac{1}{2} x+\frac{1}{8} \sin 8 x+C$
D) $\frac{1}{3} \cos ^{3} 4 x+C$

Question 5

Integrate $\int \frac{2}{x^{2}+4} d x$

A) $2 \operatorname{Arctan} x+C$
B) $2 \ln \left(x^{2}+4\right)+C$
C) $\operatorname{Arctan} \frac{x}{2}+C$
D) $\operatorname{Arctan} 2 x+C$

Question 6

Find the partial fractions of the expression $\frac{5 x^{2}+4 x+4}{x^{3}+2 x^{2}}$ .

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

Question 7

Find the partial fractions of each expression.

- $\frac{4 x+36}{(x-3)(x+5)}$

Question 8

Find the partial fractions of each expression.

- $\frac{x^{3}+33 x^{2}-12 x+108}{x^{4}-16}$

Question 9

Find the partial fractions of each expression.

- $\frac{8 \mathrm{x}+10}{(2 \mathrm{x}+5)^{2}}$

Question 10

Integrate $\int \frac{7 x-4}{(x-1)^{2}(x+2)} d x$

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

Question 11

Integrate $\int x^{2} \sin x d x$

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

Question 12

Integrate $\int \frac{\mathrm{dx}}{\left[\sqrt{\mathrm{x}^{2}+9}\right]^{3}}$

A) $\frac{x}{9 \sqrt{x^{2}+9}}+C$
B) $\frac{2}{\sqrt{\mathrm{x}^{2}+9}}+\mathrm{C}$
C) $\frac{\mathrm{x}}{\sqrt{\mathrm{x}^{2}+9}}+C$
D) $\frac{2}{3 \sqrt{\mathrm{x}^{2}+9}}+C$

Question 13

Use the following data and Simpson's Rule to approximate the area under the curve.

$\begin{array}{l|rrrrrrr} \mathrm{x} & -4 & 0 & 4 & 8 & 12 & 16 & 20 \\\hline\mathrm{y}& 11 & 9 & 3 & 10 & 21 & 30 & 40\end{array}$

A) 295
B) 262
C) 260
D) 393

Question 14

Find the area inside the polar curve $r=2-2 \sin \theta$ .

A) $6 \pi$
B) $3 \pi$
C) $12 \pi$
D) $9 \pi$

Question 15

For the problems below, integrate.

- $\int\left(x^{3}+2 x^{2}\right)^{8}\left(3 x^{2}+4 x\right) d x$

Question 16

For the problems below, integrate.

- $\int e^{6 x} d x$

Question 17

For the problems below, integrate.

- $\int \sin 5 x \cos 5 x d x$

Question 18

For the problems below, integrate.

- $\int \frac{(x+1)}{3 x^{2}+6 x} d x$

Question 19

For the problems below, integrate.

- $\int \mathrm{x}^{2} \mathrm{e}^{3} d x$

Question 20

For the problems below, integrate.
-Evaluate $\int_{2}^{4} \frac{6}{3 x-5} d x$ .

Question 21

For the problems below, integrate.

- $\int \sec ^{2}(5 x+10) d x$

Question 22

For the problems below, integrate.

- $\int \sec 3 x \tan 3 x d x$

Question 23

For the problems below, integrate.

- $\int \sin ^{3} 5 x \cos ^{2} 5 x d x$

Question 24

For the problems below, integrate.

- $\int \tan ^{3} 8 x d x$

Question 25

For the problems below, integrate.

- $\int \frac{d x}{\sqrt{5-45 x^{2}}}$

Question 26

For the problems below, integrate.

- $\int \frac{d x}{4+25 x^{2}}$

Question 27

For the problems below, write each expression as a sum of partial fractions and integrate.

- $\int \frac{(8 x-29)}{(x+2)(x-7)} d x$

Question 28

For the problems below, write each expression as a sum of partial fractions and integrate.

- $\int \frac{\left(11 x^{2}-18 x+3\right)}{x(x-1)^{2}} d x$

Question 29

For the problems below, integrate.

- $\int x e^{10 x} d x$

Question 30

For the problems below, integrate.

- $\int x \ln x d x$

Question 31

For the problems below, integrate.

- $\int \frac{d x}{\left(49-x^{2}\right)^{3 / 2}}$

Question 32

For the problems below, integrate.

- $\int \frac{\mathrm{x}^{3} \mathrm{dx}}{\sqrt{\mathrm{x}^{2}+25}}$

Question 33

For the problems below, integrate using a table or a graphing utility.

- $\int \frac{d x}{4 x(3+4 x)}$

Question 34

For the problems below, integrate using a table or a graphing utility.

- $\int e^{2 x} \sin 5 x d x$

Question 35

For the problems below, integrate using a table or a graphing utility.

-Use Simpson's rule with $n=6$ to find the approximate value of $\int_{1}^{4} \frac{d x}{x+4}$ .

Question 36

For the problems below, integrate using a table or a graphing utility.

-Find the area within the inner loop of $\mathrm{r}=4+8 \sin \theta$ .