Free Printable BC Unit 6 Review

1.

$\int_0^1\frac{5x+8}{x^2+3x+2}dx$

a)

$\ln8$

b)

$\ln\frac{27}{2}$

c)

$\ln18$

d)

$\ln288$

2.

$\int_1^e\frac{x^2+1}{x}dx$

a)

$\frac{e^2-1}{1}$

b)

$\frac{e^2+1}{2}$

c)

$\frac{e^2+2}{2}$

d)

$\frac{e^2-1}{e^2}$

3.

$\int_{-1}^4x\left(x^2-3\right)^5dx$ is equivalent to which of the following?

a)

$\frac{1}{2}\int_{-2}^{13}u^5du$

b)

$2\int_{-2}^{13}u^5du$

c)

$\int_{-2}^{13}u^5du$

d)

$\int_{-1}^4u^5du$

e)

$\frac{1}{2}\int_{-1}^4u^5du$

4.

The graph of a differentiable function f is shown. If $h\left(x\right)=\int_0^xf\left(t\right)dt$ , which of these following is true?

a)

$h\left(6\right)<h'\left(6\right)<h"\left(6\right)$

b)

$h\left(6\right)<h"\left(6\right)<h'\left(6\right)$

c)

$h'\left(6\right)<h\left(6\right)<h"\left(6\right)$

d)

$h"\left(6\right)<h\left(6\right)<h'\left(6\right)$

e)

$h"\left(6\right)<h'\left(6\right)<h\left(6\right)$

5.

Let f be a differentiable function such that

\int_{ }^{ }f\left(x\right)\sin xdx=-f\left(x\right)\cos x+\int_{ }^{ }4x^3\cos xdx

. Which of the following could be f(x)?

a)

$\cos x$

b)

$\sin x$

c)

$4x^3$

d)

$-x^4$

e)

$x^4$

6.

The graph of f', the derivative of f, consists of two line segments and a semicircle, as shown. If f(2)=1, then f(-5)=

a)

$2\pi-2$

b)

$2\pi-3$

c)

$2\pi-5$

d)

$6-2\pi$

e)

$4-2\pi$

7.

If $\int_1^xf\left(t\right)dt=\frac{20x}{\sqrt{4x^2+21}}-4$ , then $\int_1^{\infty}f\left(t\right)dt=$

a)

6

b)

1

c)

-3

d)

-4

e)

divergent

8.

If $f'\left(x\right)>0$ for all real numbers x and $\int_4^7f\left(t\right)dt=0$ , which of the following could be a table of values for the function f?

a)

b)

c)

d)

e)

9.

$\int_{ }^{ }xe^{x^2}dx=$

a)

$\frac{1}{2}e^{x^2}+C$

b)

$e^{x^2}+C$

c)

$xe^{x^2}+C$

d)

$\frac{1}{2}e^{2x}+C$

e)

$e^{2x}+C$

10.

The graph of the piecewise linear function f is shown. What is the value of $\int_{-1}^9\left(3f\left(x\right)+2\right)dx$ ?

a)

7.5

b)

9.5

c)

27.5

d)

47

e)

48.5

11.

$\int_{ }^{ }5x\left(\sqrt{x}-x^2\right)dx$

a)

$\frac{15\sqrt{x}}{2}-15x^2+C$

b)

$5x+\frac{5x^4}{4}+C$

c)

$2x^{\frac{5}{2}}-\frac{5x^4}{4}+C$

d)

$\frac{25x^{\frac{5}{2}}}{2}-\frac{5x^4}{4}+C$

e)

$\frac{5x^{\frac{7}{2}}}{3}-\frac{5x^6}{6}+C$

12.

$\int_{ }^{ }\frac{7x}{\left(2x-3\right)\left(x+2\right)}dx$

a)

$\frac{3}{2}\ln\left|2x-3\right|+2\ln\left|x+2\right|+C$

b)

$3\ln\left|2x-3\right|+2\ln\left|x+2\right|+C$

c)

$3\ln\left|2x-3\right|-2\ln\left|x+2\right|+C$

d)

$-\frac{6}{\left(2x-3\right)^2}-\frac{2}{\left(x+2\right)^2}+C$

e)

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

13.

$\int_{ }^{ }\left(3x+1\right)^5dx$

a)

$\frac{\left(3x+1\right)^6}{18}+C$

b)

$\frac{\left(3x+1\right)^6}{6}+C$

c)

$\frac{\left(3x+1\right)^6}{2}+C$

d)

$\frac{\left(\frac{3x^2}{2}+x\right)^6}{2}+C$

e)

$\left(\frac{3x^2}{2}+x\right)^5+C$

14.

$\int_{ }^{ }\frac{1}{x^2-6x+8}dx$

a)

$\ln\left|\left(x-2\right)\left(x-4\right)\right|+C$

b)

$\frac{1}{2}\ln\left|\left(x-2\right)\left(x-4\right)\right|+C$

c)

$\frac{1}{2}\ln\left|\frac{x-4}{x-2}\right|+C$

d)

$\frac{1}{2}\ln\left|\frac{x-2}{x-4}\right|+C$

15.

$\int_0^1x\sqrt{1+3x^2}dx$

a)

7/9

b)

1/6

c)

2

d)

7

16.

$\int_1^{\infty}\frac{x^2}{\left(x^3-3\right)^3}dx$

a)

$\frac{1}{24}$

b)

$-\frac{1}{24}$

c)

$\frac{1}{8}$

d)

divergent

17.

$\int_1^2\frac{x-4}{x^2}dx$

a)

-1/2

b)

ln2-2

c)

ln2

d)

2

e)

ln2+2

18.

$\int_{-2}^1\frac{\left|x\right|}{x}dx$

a)

-3

b)

-1

c)

2

d)

3

e)

nonexistent

19.

$\int_{ }^{ }\left(-3x\cos4x\right)dx$

a)

$-\frac{3}{16}\cos4x-\frac{3}{4}x\sin4x+C$

b)

$-\frac{3}{4}x\cos4x-\frac{3}{16}\sin4x+C$

c)

$-3\cos4x-3x\sin4x+C$

d)

$-\frac{3}{16}x\cos4x+\frac{3}{4}\sin4x+C$

20.

$\int_{ }^{ }x^3\ln xdx$

a)

$\frac{x^4}{4}\ln x-\frac{x^4}{4}+C$

b)

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

c)

$\frac{x^4}{4}\ln x-\frac{x^4}{16}+C$

d)

$\frac{x^4}{4}\ln x+\frac{x^4}{16}+C$

21.

$\int_{ }^{ }\frac{1}{x^2-2x}dx$

a)

$\ln\left|x\left(x-2\right)\right|+C$

b)

$\ln\left|\frac{x-2}{x}\right|+C$

c)

$\frac{1}{2}\ln\left|\frac{x-2}{x}\right|+C$

d)

$\frac{1}{2}\ln\left|\frac{x}{x-2}\right|+C$

22.

$\int_8^{\infty}\frac{1}{x^{\frac{4}{3}}}dx$

a)

-3/2

b)

-1/2

c)

1/2

d)

3/2

e)

divergent

23.

The graph of f(x) is given and consists of three line segments. Find $\int_1^4xf'\left(x\right)dx$ .

a)

5.5

b)

14.5

c)

23

d)

25

24.

$\int_1^5\frac{1}{x-3}dx$

a)

0

b)

ln2

c)

2ln2

d)

diverges

25.

$\int_{ }^{ }\frac{x}{3}e^{-2x}dx$

a)

$-\frac{1}{6}xe^{-2x}-\frac{1}{12}e^{-2x}+C$

b)

$\frac{1}{3}xe^{-2x}-\frac{1}{3}e^{-2x}+C$

c)

$\frac{1}{6}xe^{-2x}-\frac{1}{12}e^{-2x}+C$

d)

$-\frac{2}{3}xe^{-2x}-\frac{4}{3}e^{-2x}+C$

26.

$\int_{ }^{ }\frac{4x-3}{\left(2x+1\right)\left(x-2\right)}dx$

a)

$2\ln\left|2x+1\right|+\ln\left|x-2\right|+C$

b)

$\ln\left|2x+1\right|+\ln\left|x-2\right|+C$

c)

$2\ln\left|2x+1\right|-\ln\left|x-2\right|+C$

d)

$\ln\left|2x+1\right|-\ln\left|x-2\right|+C$

27.

Given k is a constant, find $\int_{ }^{ }\frac{x}{x^2+k}dx$ in terms of x and k.

a)

$\sqrt{k}\arctan\left(\frac{x}{\sqrt{k}}\right)+C$

b)

$\arctan\left(\frac{x}{\sqrt{k}}\right)+C$

c)

$\frac{1}{2}\ln\left|x^2+k\right|+C$

d)

$\ln\left|x^2+k\right|+C$

28.

$\int_{\sqrt{3}}^{\infty}\frac{1}{x^2+9}dx$

a)

$\frac{\pi}{9}$

b)

$\frac{\pi}{6}$

c)

$\frac{1}{2}\ln18$

d)

$\ln18$

e)

diverges

29.

$\int_{ }^{ }\frac{x^2}{1+x^2}dx$

a)

$x-\arctan x+C$

b)

$x+\ln\left|x^2+1\right|+C$

c)

$x+\arctan x+C$

d)

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

30.

$\int_{ }^{ }\frac{1}{x^2+4x+8}dx$

a)

$\frac{1}{2x+4}\ln\left|x^2+4x+8\right|+C$

b)

$\frac{1}{2}\arctan\left(\frac{x+2}{2}\right)+C$

c)

$\arctan\left(\frac{x+2}{2}\right)+C$

d)

$\ln\left|x^2+4x+8\right|+C$

31.

$\int_{ }^{ }\frac{1}{\sqrt{9-4x^2}}dx$

a)

$\frac{1}{2}\arcsin\left(\frac{2x}{3}\right)+C$

b)

$\frac{3}{2}\arcsin\left(\frac{2x}{3}\right)+C$

c)

$-\frac{1}{4}\sqrt{9-4x^2}+C$

d)

$-16\sqrt{9-4x^2}+C$

32.

$\int_{ }^{ }\frac{x}{\sqrt{9-4x^2}}dx$

a)

$\frac{1}{2}\arcsin\left(\frac{2x}{3}\right)+C$

b)

$\frac{3}{2}\arcsin\left(\frac{2x}{3}\right)+C$

c)

$-\frac{1}{4}\sqrt{9-4x^2}+C$

d)

$-16\sqrt{9-4x^2}+C$

33.

$\int_{ }^{ }\frac{x^2-3x+2}{x+1}dx$

a)

$\frac{1}{2}x^2-4x+6\ln\left|x+1\right|+C$

b)

$\frac{1}{2}x^2-3x+2\ln\left|x+1\right|+C$

c)

$\frac{1}{2}x^2-2x+4\ln\left|x+1\right|+C$

d)

$\frac{1}{2}x^2-2x+C$

34.

$\int_1^{\infty}\frac{1}{x}dx$

a)

divergent

b)

$\pi$

c)

e

d)

$\ln\infty$

35.

Let h be a continuous function. Using the substitution $u=2-4x$ , the integral $\int_{-3}^{-1}h\left(2-4x\right)dx$ is equal to which of the following?

a)

$\frac{1}{4}\int_6^{14}h\left(u\right)du$

b)

$-\frac{1}{4}\int_6^{14}h\left(u\right)du$

c)

$-\frac{1}{4}\int_{-3}^{-1}h\left(u\right)du$

d)

$\int_{-3}^{-1}h\left(u\right)du$

36.

\int_{ }^{ }\left[\left(x+2\right)\sqrt{x-4}\right]dx=

a)

$\frac{2}{5}\left(x-4\right)^{\frac{5}{2}}+4\left(x-4\right)^{\frac{3}{2}}+C$

b)

$\frac{2}{5}\left(x-4\right)^{\frac{5}{2}}+8\left(x-4\right)^{\frac{3}{2}}+C$

c)

$\frac{4}{5}\left(x-4\right)^{\frac{5}{2}}+\frac{2}{3}\left(x-4\right)^{\frac{3}{2}}+C$

d)

$\frac{2}{3}\left(x-4\right)^{\frac{3}{2}}+12\left(x-4\right)^{\frac{1}{2}}+C$

37.

Evaluate: $\int_2^∞\frac{dx}{x^{2\ }}$

a)

$\frac{1}{2}$

b)

$\ln2$

c)

1

d)

3

e)

divergent

38.

$\int_4^∞\frac{-2x}{\left(9-x^2\right)^{\frac{1}{3}}}dx$

a)

divergent

b)

$\left(\frac{3}{2}\right)^{\frac{2}{3}}$

c)

$9^{\frac{2}{3}}+7^{\frac{2}{3}}$

d)

$\frac{3}{2}\left(9^{\frac{2}{3}}+7^{\frac{2}{3}}\right)$

39.

$\int_0^∞x^2e^{-x^3}dx$

a)

$-\frac{1}{3}$

b)

1

c)

$\frac{1}{3}$

d)

divergent

40.

$\int_1^4\frac{x^2-2x-4}{x+1}dx$

a)

$-\frac{4}{7}$

b)

$\frac{20}{9}$

c)

$-\frac{3}{2}+\ln\frac{2}{5}$

d)

$\frac{9}{2}+5\ln\frac{2}{5}$

41.

$\int_{ }^{ }\frac{1}{x^2+3x-10}dx$

a)

$\frac{1}{7}\ln\left|\frac{x-2}{x+5}\right|+C$

b)

$\frac{1}{7}\ln\left|\frac{x+2}{x-5}\right|+C$

c)

$\frac{1}{7}\ln\left|\frac{x-5}{x+2}\right|+C$

d)

$\frac{1}{7}\ln\left|\frac{x+2}{x-5}\right|+C$

42.

$\int3x\ e^{2x}\ dx\$

a)

$-\frac{xe^{2x}}{2}+\frac{e^{2x}}{4}+C$

b)

$\frac{3xe^{2x}}{2}-\frac{3e^{2x}}{4}+C$

c)

$xe^{-2x}+\frac{\left(1-x^2\right)^{ }}{2}+C$

d)

$-\frac{xe^{2x}}{2}+\frac{\ln e^{2x}}{4}+C$

43.

$\int\frac{dx}{x^2+2x-15}$

a)

$\frac{1}{8}\ln\left|x-3\right|-\frac{1}{8}\ln\left|x+5\right|+c$

b)

$\frac{1}{4}\ln\left|x-3\right|-\frac{1}{8}\ln\left|x+5\right|+c$

c)

$\frac{1}{8}\ln\left|x+5\right|-\frac{1}{8}\ln\left|x-3\right|+c$

d)

$\frac{1}{4}\ln\left|x+5\right|-\frac{1}{4}\ln\left|x-3\right|+c$

44.

$\int_0^1\left(x-1\right)\sin\left(\pi x\right)dx=$

a)

$-1$

b)

$0$

c)

$-\frac{1}{\pi}$

d)

$\pi$

e)

$-\pi$

45.

Evaluate $\int_{-\infty}^{\infty}\frac{x}{\left(x^2+3\right)^2}dx$

a)

diverges

b)

0

c)

-2

d)

4

46.

Which of these integrals are improper integrals?

a)

$\int_{-\infty}^1\ x^{4\ }dx$

b)

$\text{\displaystyle}\int_{-7}^5\ \frac{x+2}{x-5}dx\$

c)

$\int_{-7}^{10}\ \frac{x+2}{x-5}dx\$

d)

$\int_{-7}^{-1}\ \frac{x+2}{x-5}dx\$

e)

$\int_{-1}^{\infty}\ x^{4\ }dx$

47.

$\int_2^5\ \frac{1}{\sqrt{5-x}}dx$

a)

$\frac{1}{\sqrt[]{3}}$

b)

$2\sqrt{3}$

c)

diverges

d)

$\sqrt{3}$

48.

$\int t\sin t\ dt\$

a)

$-t\cos t+\sin t+C$

b)

$t\cos t+\sin t+C$

c)

$t\cos t-\sin t+C$

d)

$-t\sin t-\cos t+C$

49.

$\int xe^{4x}dx$

a)

$4xe^{4x}+16e^{4x}+C$

b)

$4xe^{4x}-16e^{4x}+C$

c)

$\frac{xe^{4x}}{4}+\frac{e^{4x}}{16}+C$

d)

$\frac{xe^{4x}}{4}-\frac{e^{4x}}{16}+C$

50.

$\int x^{4\ }\ln x\ dx$

a)

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

b)

$\frac{2x^{\frac{3}{2}}\ln4x}{3}-\frac{4x^{\frac{3}{2}}}{9}+C$

c)

$\frac{\left(4x^2-1\right)\cdot e^{4x^2}}{32}+C$

d)

$\frac{x^5\ln x}{5}-\frac{x^5}{25}+C$

51.

What is the formula for Integration by parts?

a)

$\int_{ }^{ }u\ dv\ =\ uv-\int_{ }^{ }v\ du$

b)

$\int_{ }^{ }v\ dv\ =\int_{ }^{ }u\ du\ -uv$

c)

$\int_{ }^{ }u\ du\ =\ vu\ -\int_{ }^{ }v\ dv$

d)

$\int_{ }^{ }u\ dv\ =\ uv+\int_{ }^{ }v\ du$

BC Unit 6 Review

Total questions: 51

Worksheet time: 4hrs 15mins

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