Deck 13: Integral Calculus

Question 1

Evaluate.

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

A) $3 x^{2}+C$
B) $4 x^{4}+C$
C) $\frac{1}{4} x^{4}+C$
D) $\frac{1}{3} x^{2}+C$

Answer 1

$\frac{1}{4} x^{4}+C$

Question 2

Evaluate.

- $\int 4 x^{2 / 3} d x$

A) $\frac{8}{3} x^{5 / 3}+$ C
B) $\frac{4}{3} x^{5 / 3}+C$
C) $\frac{12}{5} x^{5 / 3}+C$
D) $6 x^{5 / 3}+C$

Answer 2

$\frac{12}{5} x^{5 / 3}+C$

Question 3

Evaluate.

- $\int 20 d x$

A) 0
B) $20+C$
C) $10 x^{2}$
D) $20 x+C$

Answer 3

$20 x+C$

Question 4

Evaluate.

- $\int 7 x^{-3} d x$

A) $7 x^{3}$
B) $-21 x+C$
C) $x+7 C$
D) $-\frac{7}{2 x^{2}}+C$

Question 5

Evaluate.

- $\int 12 x^{3} \sqrt{x} d x$

A) $\frac{11}{5} x^{9 / 2}+C$
B) $\frac{8}{3} x^{9 / 2}+C$
C) $\frac{2}{9} x^{9 / 2}+C$
D) $\frac{24}{7} x^{9 / 2}+C$

Question 6

Evaluate.

- $\int 9 \sqrt[3]{x} d x$

A) $27 x^{4 / 3}+C$
B) $\frac{27}{4} x^{4 / 3}+C$
C) $\frac{9}{4} x^{4 / 3}+C$
D) $9 x^{4 / 3}+C$

Question 7

Evaluate.

- $\int \frac{28}{x^{2}} d x$

A) $-28 x+C$
B) $28 x+C$
C) $-\frac{28}{x}+C$
D) $\frac{28}{x}+C$

Question 8

Evaluate.

- $\int 15 x^{-8} d x$

A) $-\frac{15}{7} x^{-7}+C$
B) $\left(\frac{105}{\mathrm{x}}\right)^{7}+C$
C) $\left(\frac{15}{7 x}\right)^{9}+C$
D) $-120 x^{-9}+C$

Question 9

Evaluate.

- $\int\left(7 x^{2}+1\right) d x$

A) $\frac{7}{3} x^{3}+x+C$
B) $\frac{7}{3} x^{3}+C$
C) $14 x+C$
D) $x+C$

Question 10

Evaluate.

- $\int\left(2 x^{2}-7 x\right) d x$

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

Question 11

Evaluate.

- $\int\left(2 x^{5}-7 x^{3}+6\right) d x$

A) $6 x^{6}-\frac{7}{3} x^{4}+6 x+C$
B) $\frac{1}{3} x^{6}-\frac{7}{4} x^{4}+6 x+C$
C) $6 x^{6}-\frac{7}{4} x^{4}+6 x+C$
D) $\frac{1}{2} x^{6}-\frac{7}{3} x^{4}+6 x+C$

Question 12

Evaluate.

- $\int\left(4 t^{2}+4 t-5\right) d t$

A) $\frac{4}{3} t^{3}+2 t^{2}-5 t+C$
B) $8 t+4+C$
C) $4 t^{3}+4 t^{2}-5 t+C$
D) $2 t^{3}+4 t^{2}-5 t+C$

Question 13

Evaluate.

- $\int\left(7 x^{2}+x^{-3}\right) d x$

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

Question 14

Evaluate.

- $\int\left(x^{4 / 3}-3 x^{5 / 2}\right) d x$

A) $\frac{3}{7} x^{7 / 3}-\frac{2}{7} x^{7 / 2}+C$
B) $\frac{3}{4} x^{7 / 3}-\frac{4}{7} x^{7 / 2}+C$
C) $\frac{3}{4} x^{7 / 3}-\frac{3}{7} x^{7 / 2}+C$
D) $\frac{3}{7} x^{7 / 3}-\frac{6}{7} x^{7 / 2}+C$

Question 15

Evaluate.

- $\int(\sqrt{x}+\sqrt[3]{x}) d x$

A) $2 \sqrt{x}+3 \sqrt[3]{x}+C$
B) $2 \sqrt{x}+2 \sqrt[3]{x}+C$
C) $\frac{2}{3} x^{3 / 2}+\frac{3}{4} x^{4 / 3}+$ C
D) $\frac{1}{2} x^{3 / 2}+\frac{2}{3} x^{4 / 3}+C$

Question 16

Evaluate.

- $\int \frac{5}{\mathrm{x}^{2}}-\frac{4}{\sqrt{\mathrm{x}}} \mathrm{dx}$

A) $-\frac{5}{x}-8 \sqrt{x}+C$
B) $-\frac{5}{x}-\frac{8}{\sqrt{x}}+C$
C) $\frac{5}{x}-8 \sqrt{x}-C$
D) $\frac{5}{x}-\frac{8}{\sqrt{x}}+C$

Question 17

Evaluate.

- $\int \sqrt{\mathrm{x}}+\frac{\mathrm{x}}{\mathrm{x}^{5}} \mathrm{dx}$

A) $2 x \sqrt{x}-\frac{1}{3} x^{-3}+C$
B) $\frac{2}{3} x^{3 / 2}-\frac{1}{4} x^{-4}+C$
C) $\frac{2}{3} x^{3 / 2}-\frac{1}{3} x^{-3}+C$
D) $2 x \sqrt{x}-\frac{1}{4} x^{-4}+C$

Question 18

Evaluate.

- $\int \frac{3 \sqrt{\mathrm{x}}-5}{\mathrm{x}^{2}} \mathrm{dx}$

A) $-\frac{6}{\sqrt{x}}+\frac{5}{x}+C$
B) $\frac{6}{\sqrt{x}}-\frac{5}{x}+C$
C) $\frac{6}{\sqrt{x}}+\frac{5}{x}+C$
D) $-\frac{6}{\sqrt{x}}-\frac{5}{x}+C$

Question 19

Evaluate.

- $\int 8 e^{4 y} d y$

A) $\frac{1}{2} e^{4 y}+C$
B) $\frac{1}{4} e^{4 y}+C$
C) $4 e^{4 y}+C$
D) $2 e^{4 y}+C$

Question 20

Evaluate.

- $\int 12 e^{-0.2 x} d x$

A) $\frac{12 e^{-0.2 x+1}}{-0.2 x+1}+C$
B) $-2.4 e^{-0.2 x}+C$
C) $12 e^{-0.2 x}+C$
D) $-60 e^{-0.2 x}+C$

Question 21

Evaluate.

- $\int\left(t^{3}+e^{2 t}\right) d t$

A) $\frac{t^{4}}{4}+\frac{e^{2 t}}{2}+C$
B) $\frac{t^{2}}{2}+2 e^{2 t}+C$
C) $\frac{\mathrm{t}^{4}}{4}+\frac{\mathrm{e}^{3 \mathrm{t}}}{3}+\mathrm{C}$
D) $\frac{t^{4}}{4}+e^{2 t}+C$

Question 22

Evaluate.

- $\int 10^{2 x} d x$

A) $\frac{10^{2 x}}{\ln 10}+C$
B) $\frac{10^{2 x}}{2}+C$
C) $\frac{10^{3 x}}{3}+C$
D) $\frac{10^{2 x}}{2 \ln 10}+C$

Question 23

Find the integral.

- $\int\left(7 x^{-4}-9 x^{-1}\right) d x$

A) $-\frac{7}{3} x^{-3}-9 \ln |x|+C$
B) $-\frac{7}{3} x^{-3}+9 \ln |x|+C$
C) $\frac{7}{4} x^{-3}+9 \ln |x|+C$
D) $\frac{7}{4} x^{-3}-9 \ln |x|+C$

Question 24

Find the integral.

- $\int\left(6 x-6 x^{-1}\right) d x$

A) $12 x^{2}-6 \ln |x|+C$
B) $3 x^{2}-6 \ln |x|+C$
C) $3 x^{2}+3 x^{-2}+C$
D) $12 x^{2}+3 x^{-2}+C$

Question 25

Find the integral.

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

A) $2 x+2 \ln \left|x^{2}\right|+3 \ln \left|x^{3}\right|+C$
B) $\ln |x|+2 \ln \left|x^{2}\right|+3 \ln \left|x^{3}\right|+C$
C) $\ln |x|-2 x^{-1}-\frac{3}{2 x^{-2}}+C$
D) $\frac{2}{x^{2}}+\frac{6}{x^{3}}+\frac{12}{x^{4}}+C$

Question 26

Find the integral.

- $\int\left(3 x+5 x^{-1}\right) d x$

A) $3 x^{3}+30 x-\frac{25}{3} x^{-1}+C$
B) $3 x^{3}+15 x-\frac{25}{3} x^{-1}+C$
C) $\frac{3}{2} x^{2}+5 \ln |x|+C$
D) $\frac{9}{4} x^{4}+25 \ln \left|x^{2}\right|+C$

Question 27

Find the integral.

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

A) $\frac{3}{2} x^{4}-\ln |x|+C$
B) $\frac{3}{4} x^{4}-\ln |x|+C$
C) $\frac{3}{4} x^{4}+\ln |x|+C$
D) $\frac{3}{2} x^{4}+\ln |x|+C$

Question 28

Find the integral.

- $\int\left(\frac{x}{6}+\frac{6}{x}\right) d x$

A) $\frac{1}{12} x^{2}+6 \ln |x|+C$
B) $\frac{1}{6} x+C$
C) $x+C$
D) $x \ln 6+6 \ln |x|+C$

Question 29

Find the integral.

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

A) $4 \ln |x|+2 e^{x}+C$
B) $4 \ln |x|+2 x e^{x-1}+C$
C) $\frac{8}{x^{2}}+2 x e^{x-1}+C$
D) $\frac{8}{x^{2}}+2 e^{x}+C$

Question 30

Find the integral.

- $\int \frac{x^{5}+1}{x} d x$

A) $\frac{1}{3} x^{4}-\ln |x|+C$
B) $\frac{1}{5} x^{5}+\ln |x|+C$
C) $\frac{1}{5} x^{5}-\ln |x|+C$
D) $\frac{1}{3} x^{4}+\ln |x|+C$

Question 31

Find the integral.

- $\int \frac{\sqrt{\mathrm{x}}-6}{2 \mathrm{x} \sqrt{\mathrm{x}}} \mathrm{dx}$

A) $\frac{1}{2} \sqrt{x} \ln |x|-6 x^{-1 / 2}+C$
B) $\frac{1}{2} \sqrt{x} \ln |x|+6 x^{-1 / 2}+C$
C) $\frac{1}{2} \ln |x|-6 x^{-1 / 2}+C$
D) $\frac{1}{2} \ln |x|+6 x^{-1 / 2}+C$

Question 32

Find the integral.

- $\int(5 x-3)^{2} d x$

A) $\frac{25}{3} x^{3}+9 x+C$
B) $\frac{5}{2} x^{2}+9 x+C$
C) $\frac{5}{2} x^{3}-15 x^{2}+9 x+C$
D) $\frac{25}{3} x^{3}-15 x^{2}+9 x+C$

Question 33

Find the integral.

- $\int x^{4}\left(x^{3}+12 x-7\right) d x$

A) $\frac{x^{5}}{5}\left(\frac{x^{4}}{4}+6 x^{2}-7 x\right)+C$
B) $\frac{x^{8}}{8}+2 x^{6}-\frac{7 x^{5}}{5}+C$
C) $\frac{x^{8}}{8}+6 x^{2}-7 x+C$
D) $\frac{x^{5}}{5}\left(x^{3}+12 x-7\right)+x^{4}\left(\frac{x^{4}}{4}+6 x^{2}-7 x\right)+C$

Question 34

Find the integral.

- $\int\left(\frac{9}{x}-8 e^{-.7 x}\right) d x$

A) $\frac{18}{x^{2}}-\frac{28}{5} e^{-.7 x}+C$
B) $-\frac{9}{x^{2}}+\frac{28}{5} e^{-.7 x}+C$
C) $9 \ln |x|-\frac{80}{7} e^{-.7 x}+C$
D) $9 \ln |x|+\frac{80}{7} e^{-.7 x}+C$

Question 35

Solve the problem.

-The slope to the tangent line of a curve is given by $f^{\prime}(x)=x^{2}-11 x+7$ . If the point $(0,5)$ is on the curve, find an equation of the curve.

A)f $(x)=\frac{1}{3} x^{3}-12 x^{2}+7 x+5$
B) $f(x)=\frac{1}{3} x^{3}-\frac{11}{2} x^{2}+7 x+5$
C) $f(x)=\frac{1}{3} x^{3}-\frac{11}{2} x^{2}+7 x+1$
D) $f(x)=\frac{1}{3} x^{3}-12 x^{2}+7 x+1$

Question 36

Solve the problem.

-Find $C(x)$ if $C^{\prime}(x)=\sqrt{x}$ and $C(9)=40$ .

A) $C(x)=\frac{2}{3} x^{3 / 2}+31$
B) $C(x)=2 x^{3 / 2}+22$
C) $C(x)=2 x^{3 / 2}+31$
D) $C(x)=\frac{2}{3} x^{3 / 2}+22$

Question 37

Solve the problem.

-Find the cost function if the marginal cost function is $C^{\prime}(x)=18 x-5$ and the fixed cost is $\$ 3$ .

A) $C(x)=18 x^{2}-5 x+2$
B) $C(x)=9 x^{2}-5 x+2$
C) $C(x)=18 x^{2}-5 x+3$
D) $C(x)=9 x^{2}-5 x+3$

Question 38

Solve the problem.

-Find $C(x)$ if $C^{\prime}(x)=5 x^{2}-7 x+4$ and $C(6)=260$ .

A) $C(x)=\frac{5}{3} x^{3}-\frac{7}{2} x^{2}+4 x-260$
B) $C(x)=\frac{5}{3} x^{3}-\frac{7}{2} x^{2}+4 x+2$
C) $C(x)=\frac{5}{3} x^{3}-\frac{7}{2} x^{2}+4 x-2$
D) $C(x)=\frac{5}{3} x^{3}-\frac{7}{2} x^{2}+4 x+260$

Question 39

Solve the problem.

-The rate at which an assembly line worker's efficiency $E$ (expressed as a percent) changes with respect to time $t$ is given by $E^{\prime}(t)=70-6 t$ , where $t$ is the number of hours since the worker's shift began. Assuming that $\mathrm{E}(1)=92$ , find $\mathrm{E}(\mathrm{t})$ .

A) $E(t)=70 t-3 t^{2}+92$
B) $E(t)=70 t-6 t^{2}+25$
C) $E(t)=70 t-3 t^{2}+25$
D) $E(t)=70 t-3 t^{2}+159$

Question 40

Solve the problem.

-Suppose that a velocity function is given by $v(t)=2 t^{4}$ . Find the position function $s(t)$ if $s(0)=7$ .

A) $s(t)=\frac{2}{5} t^{5}+7$
B) $s(t)=2 t^{5}+7$
C) $s(t)=\frac{2}{5} t^{5}$
D) $s(t)=8 t^{3}+7$

Question 41

Solve the problem.

-Suppose that an object's acceleration function is given by $a(t)=10 t+6$ . The object's initial velocity, $v(0)$ , is 3 , and the object's initial position, $s(0)$ , is 12 . Find $s(t)$ .

A) $\frac{10}{3} t^{3}+3 t^{2}+12 t+3$
B) $\frac{5}{3} t^{3}+3 t^{2}+3 t+12$
C) $5 t^{2}+6 t+3$
D) $\frac{5}{3} t^{3}+3 t^{2}+3 t$

Question 42

Solve the problem.

-Suppose that the acceleration of an object is given by $a(t)=5 t^{2 / 3}+2 e^{-t}$ . The velocity of the object at time 0 is $v(0)=5$ and its position at time 0 is $s(0)=-9$ . Find $s(t)$ .

A) $\frac{9 t^{8 / 3}}{8}+2 e^{-t}+5 t-9$
B) $\frac{9 t^{8 / 3}}{8}+2 e^{-t}+7 t-11$
C) $\frac{200 t^{8 / 3}}{9}+2 e^{-t}+3 t-11$
D) $3 t^{5 / 3}-2 e^{-t}+7$

Question 43

Find the integral.

- $\int 4(2 x+5)^{3} d x$

A) $\frac{3}{4}(2 x+5)^{4}+$ C
B) $\frac{1}{2}(2 x+5)^{4}+C$
C) $\frac{3}{8}(2 x+5)^{4}+C$
D) $\frac{1}{4}(2 x+5)^{4}+C$

Question 44

Find the integral.

- $\int 3(2 t+5)^{3} d t$

A) $\frac{1}{4}(2 t+5)^{4}+C$
B) $\frac{3}{8}(2 t+5)^{4}+C$
C) $\frac{1}{2}(2 t+5)^{4}+C$
D) $\frac{3}{4}(2 t+5)^{4}+C$

Question 45

Find the integral.

- $\int \frac{8}{(y-9)^{3}} d y$

A) $\frac{2}{(y-9)^{4}}+C$
B) $\frac{-2}{(y-9)^{4}}+C$
C) $\frac{4}{(y-9)^{2}}+C$
D) $\frac{-4}{(y-9)^{2}}+C$

Question 46

Find the integral.

- $\int \sqrt{6 x-7} d x$

A) $\frac{1}{2}(6 x-7)^{3 / 2}+C$
B) $\frac{1}{9}(6 x-7)^{3 / 2}+C$
C) $\frac{1}{3}(6 x-7)^{3 / 2}+C$
D) $\frac{1}{6}(6 x-7)^{3 / 2}+C$

Question 47

Find the integral.

- $\int \frac{\mathrm{dr}}{\sqrt{6 \mathrm{r}-7}}$

A) $\frac{1}{3} \sqrt{6 r-7}+C$
B) $\frac{1}{4} \sqrt{6 r-7}+C$
C) $\frac{1}{6} \sqrt{6 r-7}+C$
D) $\frac{1}{2} \sqrt{6 r-7}+C$

Question 48

Find the integral.

- $\int 9 z \sqrt{3 z^{2}-7} d z$

A) $\frac{1}{2} z\left(3 z^{2}-7\right)^{3 / 2}+C$
B) $\left(3 z^{2}-7\right)^{3 / 2}+C$
C) $z\left(3 z^{2}-7\right)^{3 / 2}+C$
D) $\frac{1}{2}\left(3 z^{2}-7\right)^{3 / 2}+C$

Question 49

Find the integral.

- $\int \frac{19}{2+5 y} d y$

A) $19 \ln |2+5 y|+C$
B) $\frac{19}{5} \ln |2+5 y|+C$
C) $18 \ln |2+5 y|+C$
D) $\frac{18}{5} \ln |2+5 y|+C$

Question 50

Find the integral.

- $\int \frac{x d x}{\left(7 x^{2}+3\right)^{5}}$

A) $-\frac{7}{3}\left(7 x^{2}+3\right)^{-4}+C$
B) $-\frac{1}{14}\left(7 x^{2}+3\right)^{-6}+C$
C) $-\frac{7}{3}\left(7 x^{2}+3\right)^{-6}+C$
D) $-\frac{1}{56}\left(7 x^{2}+3\right)^{-4}+C$

Question 51

Find the integral.

- $\int(y-1)^{7} d y$

A) $\frac{1}{2} y^{2}+\frac{1}{8}(y-1)^{8}+C$
B) $\frac{1}{16(y-1)^{8}}+C$
C) $\frac{1}{9}(y-1)^{9}+\frac{1}{8}(y-1)^{8}+C$
D) $\frac{1}{8}(y-1)^{8}+C$

Question 52

Find the integral.

- $\int \frac{\mathrm{e}^{\mathrm{x}} \mathrm{dx}}{\mathrm{e}^{\mathrm{x}}+\mathrm{e}}$

A) $x+C$
B) $\frac{x}{e}+C$
C) $e \ln \left|e^{x}+e\right|+C$
D) $\ln \left|e^{x}+e\right|+C$

Question 53

Find the integral.

- $\int 8 e^{4 y} d y$

A) $4 e^{4 y}+C$
B) $\frac{1}{4} e^{4 y}+C$
C) $2 e^{4 y}+C$
D) $\frac{1}{2} e^{4 y}+C$

Question 54

Find the integral.

- $\int 2 e^{4 z} d z$

A) $\frac{1}{2} e^{4 z}+C$
B) $\frac{1}{4} e^{4 z}+C$
C) $4 e^{4 z}+C$
D) $2 e^{4 z}+C$

Question 55

Find the integral.

- $\int 16 p^{2} e^{\left(4 p^{3}\right)} d p$

A) $4 e^{\left(4 p^{3}\right)}+C$
B) $\frac{16}{3} e^{\left(4 p^{3}\right)}+C$
C) $\frac{4}{3} e^{\left(4 p^{3}\right)}+C$
D) $\frac{16}{9} e^{\left(4 p^{3}\right)}+C$

Question 56

Find the integral.

- $\int t e^{\left(-7 t^{2}\right)} d t$

A) $-\frac{1}{7} e^{\left(-7 t^{2}\right)}+C$
B) $\frac{1}{7} e^{\left(-7 t^{2}\right)}+C$
C) $-\frac{1}{14} \mathrm{e}^{\left(-7 \mathrm{t}^{2}\right)}+\mathrm{C}$
D) $\frac{1}{14} e^{\left(-7 t^{2}\right)}+C$

Question 57

Find the integral.

- $\int(1-6 x) e^{\left(3 x-9 x^{2}\right)} d x$

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

Question 58

Find the integral.

- $\int \frac{5 e^{1 / y}}{3 y^{2}} d y$

A) $-\frac{5}{3} e^{1 / y}+C$
B) $\frac{5 e^{1 / y}}{y^{3}}+C$
C) $\frac{5}{3} \mathrm{e}^{1 / y+C}$
D) $\frac{5 e^{1 / y}}{3} 6 y+C$

Question 59

Find the integral.

- $\int \frac{3 \mathrm{e}^{\sqrt{\mathrm{z}}}}{8 \sqrt{\mathrm{z}}} \mathrm{dz}$

A) $-12 e^{\sqrt{z}}+C$
B) $-24 e^{\sqrt{z}}+C$
C) $\frac{3}{4} \mathrm{e}^{\sqrt{\mathrm{z}}}+\mathrm{C}$
D) $\frac{3}{8} e^{\sqrt{z}}+C$

Question 60

Find the integral.

- $\int 7 p e^{\left(5 p^{2}\right)} d p$

A) $7 e^{\left(5 p^{2}\right)}+C$
B) $-7 e^{\left(5 p^{2}\right)}+C$
C) $\frac{7}{10} e^{\left(5 p^{2}\right)}+C$
D) $-\frac{7}{5} e^{\left(5 p^{2}\right)}+C$

Question 61

Find the integral.

- $\int 11 x^{2} e^{\left(-4 x^{3}\right)} d x$

A) $\frac{11}{12} e^{\left(-4 x^{3}\right)}+C$
B) $-11 e^{\left(-4 x^{3}\right)}+C$
C) $12 e^{\left(-4 x^{3}\right)}+C$
D) $-\frac{11}{12} e^{\left(-4 x^{3}\right)}+C$

Question 62

Find the integral.

- $\int \frac{19}{2+5 y} d y$

A) $\frac{19}{5} \ln |2+5 y|+C$
B) $18 \ln |2+5 y|+C$
C) $\frac{18}{5} \ln |2+5 y|+C$
D) $19 \ln |2+5 y|+C$

Question 63

Find the integral.

- $\int \frac{e^{x} d x}{e^{x}+e}$

A) $\frac{x}{e}+C$
B) $\ln \left|e^{x}+e\right|+C$
C) $x+C$
D) $e \ln \left|e^{x}+e\right|+C$

Question 64

Find the integral.

- $\int \frac{3}{3 x+1} d x$

A) $3 \ln |3 x+1|+C$
B) $\ln (3 x+1)+C$
C) $\ln |3 x+1|+C$
D) $\frac{1}{3} \ln |3 x+1|+C$

Question 65

Find the integral.

- $\int \frac{x^{5}}{x^{6}+7} d x$

A) $\frac{1}{5} \ln \left|x^{6}+7\right|+C$
B) $-\ln \left(x^{6}+7\right)+C$
C) $\frac{1}{6} \ln \left|x^{6}+7\right|+C$
D) $6 \ln \left|x^{6}+7\right|+C$

Question 66

Find the integral.

- $\int \frac{1}{8-x} d x$

A) $-\ln |8-x|+C$
B) $\ln (8-x)+C$
C) $\ln |8-x|+C$
D) $-\frac{1}{8} \ln |8-x|+C$

Question 67

Find the integral.

- $\int \frac{x^{4}}{5 x^{5}+3} d x$

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

Question 68

Find the integral.

- $\int \frac{d x}{(5+\ln x) x}$

A) $\ln |5+\ln x|+C$
B) $\ln (5+\ln x)+C$
C) $5 \ln |5+\ln x|+C$
D) $\frac{1}{5} \ln |5+\ln x|+C$

Question 69

Find the integral.

- $\int \frac{x}{\sqrt{2 x-10}} d x$

A) $\frac{2}{4}\left(\frac{x^{3}}{3}+10 x\right)+C$
B) $\frac{1}{6}(2 x-10)^{3 / 2}+\frac{10}{2}(2 x-10)^{1 / 2}+C$
C) $\frac{1}{6}(2 x-10)^{3 / 2}+5(2 x-10)^{1 / 2}+C$
D) $\frac{2}{12}\left(\frac{x^{3}}{3}+10 x\right)+C$

Question 70

Find the integral.

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

A) $\frac{1}{3}\left[\frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}\right]+C$
B) $\frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}+C$
C) $\frac{1}{3} \frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}+C$
D) $\frac{1}{3}\left[\frac{(3 x+1)^{5 / 3}}{2}-\frac{(3 x+1)^{2 / 3}}{5}\right]+C$

Question 71

Find the integral.

- $\int \frac{x}{(1-x)^{4}} d x$

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

Question 72

Find the integral.

- $\int \frac{e^{x} d x}{e^{x}+e}$

A) $e \ln \left|e^{x}+e\right|+C$
B) $x+C$
C) $\ln \left|e^{x}+e\right|+C$
D) $\frac{x}{e^{x}+C}$

Question 73

Solve the problem.

-The rate of expenditure of a particular machine is given by $M^{\prime}(x)=15 x \sqrt{x^{2}+5}$ , where $x$ is time measured in years. Maintenance costs through the second year are $\$ 125$ . Find the total maintenance function.

A)M $(x)=5\left(x^{2}+5\right)^{3 / 2}+110$
B) $M(x)=15\left(x^{2}+5\right)^{3 / 2}-10$
C)M $(\mathrm{x})=15\left(\mathrm{x}^{2}+5\right)^{3 / 2}+110$
D) $M(x)=5\left(x^{2}+5\right)^{3 / 2}-10$

Question 74

Solve the problem.

-The rate of growth of profit (in millions) from an invention is approximated by $P^{\prime}(x)=x e^{-x^{2}}$ , where $x$ represents time in years. The total profit in year 1 that the invention is in operation is $\$$ 45,000 . Find the total profit function $\mathrm{P}(\mathrm{x})$ .

A) $P(x)=-\frac{1}{2} e^{-x^{2}}-229,000$
B) $P(x)=-.5 e^{-x^{2}}+0.229$
C) $P(x)=-.5 e^{-x^{2}}+229,000$
D) $P(x)=-\frac{1}{2} e^{-x^{2}}-0.229$

Question 75

Solve the problem.

-The second derivative of a person's body temperature, with respect to the dosage of $x$ milligrams of a drug, is given by $D^{\prime}(x)=\frac{8}{x+5}$ . One milligram raises the temperature $1.5^{\circ} \mathrm{C}$ . Find the function $D(x)$ giving the total change in temperature as a function of $x$ .

A) $\mathrm{D}(\mathrm{x})=\ln \left|\frac{8}{\mathrm{x}+5}\right|+12.8$
B) $\mathrm{D}(\mathrm{x})=8 \ln |\mathrm{x}+5|-12.8$
C) $D(x)=8 \ln |x+5|+1.5$
D) $\mathrm{D}(\mathrm{x})=\ln \left|\frac{8}{\mathrm{x}+5}\right|-1.5$

Question 76

Solve the problem.

-The velocity $\mathrm{v}$ (in $\mathrm{m} / \mathrm{s}$ ) of an object moving with acceleration a (in $\mathrm{m} / \mathrm{s}^{2}$ ) is given by $\mathrm{v}=\int \mathrm{adt}$ , where $t$ is the time (in seconds). Find a formula for $v$ , if $a=\frac{5}{4} t\left(t^{2}+1\right)^{2}$ .

A) $v=\frac{1}{2} \frac{\left(t^{2}+1\right)^{3}}{3}+C$
B) $v=\frac{5}{8} \frac{\left(t^{2}+1\right)^{3}}{3}+C$
C) $\mathrm{v}=\frac{1}{4} \frac{\left(\mathrm{t}^{2}+1\right)^{3}}{3}+\mathrm{C}$
D) $\mathrm{v}=-\frac{1}{8} \frac{\left(\mathrm{t}^{2}+1\right)^{3}}{3}+\mathrm{C}$

Question 77

Solve the problem.

-The work $W$ (in joules) done by a force $F$ (in newtons) moving an object through a distance $x$ (in meters) is given by $\mathrm{W}=\int \mathrm{Fdx}$ . Find a formula for $\mathrm{W}$ , if $\mathrm{F}=\mathrm{kx}$ and $\mathrm{k}$ is a constant.

A) $W=\frac{k x}{2}+C$
B) $\mathrm{W}=\frac{k x^{2}}{2}+C$
C) $\mathrm{W}=\mathrm{kx} \mathrm{x}^{2}+\mathrm{C}$
D) $\mathrm{W}=\mathrm{k}+\mathrm{C}$

Question 78

Solve the problem.

-The current (in amperes) in an inductor of inductance $\mathrm{L}$ (in henries) is given by $\mathrm{i}=\frac{1}{\mathrm{~L}} \int \mathrm{Vdt}$ , where $\mathrm{V}$ is the voltage (in volts) and $\mathrm{t}$ is the time (in seconds). Find a formula for $\mathrm{i}$ , if $\mathrm{V}=7 \mathrm{t}\left(\mathrm{t}^{2}-4\right)$ .

A) $\mathrm{i}=\mathrm{L}\left(\frac{7}{4}\left(\mathrm{t}^{2}-4\right)\right)+\mathrm{C}$
B) $i=\mathrm{L}\left(\frac{7}{4}\left(t^{2}-4\right)^{2}\right)+C$
C) $i=\frac{1}{L}\left(\frac{7}{4}\left(t^{2}-4\right)^{2}\right)+C$
D) $\mathrm{i}=\frac{1}{\mathrm{~L}}\left(\frac{7}{4}\left(\mathrm{t}^{2}-4\right)\right)+\mathrm{C}$

Question 79

Use integration by parts to find the integral.

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

A) $3 x e^{x}-3 e^{x}+C$
B) $3 e^{x}-e^{x}+C$
C) $3 e^{x}-3 x e^{x}+C$
D) $x e^{x}-3 e^{x}+C$

Question 80

Use integration by parts to find the integral.

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

A) $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{4} x e^{2} x+\frac{1}{4} e^{2 x}+C$
B) $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{2} x e^{2 x}+\frac{1}{4} e^{2 x}+C$
C) $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{2} x e^{2 x}+C$
D) $\frac{1}{2} x^{2} e^{2 x}-x e^{2 x}+\frac{1}{4} e^{2 x}+C$