5.3E: Exercises for Section 5.3

In exercises 1 - 20, find the antiderivative $F(x)$ of each function $f(x).$

1) $f(x)=\dfrac{1}{x^2}+x$

2) $f(x)=e^x−3x^2+\sin x$

Answer

$F(x)=e^x−x^3−\cos x+C$

3) $f(x)=e^x+3x−x^2$

4) $f(x)=x−1+4\sin(2x)$

Answer

$F(x)=\dfrac{x^2}{2}−x−2\cos(2x)+C$

5) $f(x)=5x^4+4x^5$

6) $f(x)=x+12x^2$

Answer

$F(x)=\frac{1}{2}x^2+4x^3+C$

7) $f(x)=\dfrac{1}{\sqrt{x}}$

8) $f(x)=\left(\sqrt{x}\right)^3$

Answer

$F(x)=\frac{2}{5}\left(\sqrt{x}\right)^5+C$

9) $f(x)=x^{1/3}+\big(2x\big)^{1/3}$

10) $f(x)=\dfrac{x^{1/3}}{x^{2/3}}$

Answer

$F(x)=\frac{3}{2}x^{2/3}+C$

11) $f(x)=2\sin(x)+\sin(2x)$

12) $f(x)=\sec^2 x +1$

Answer

$F(x)=x+\tan x+C$

13) $f(x)=\sin x\cos x$

14) $f(x)=\sin^2(x)\cos(x)$

Answer

$F(x)=\frac{1}{3}\sin^3(x)+C$

15) $f(x)=0$

16) $f(x)=\frac{1}{2}\csc^2 x+\dfrac{1}{x^2}$

Answer

$F(x)=−\frac{1}{2}\cot x −\dfrac{1}{x}+C$

17) $f(x)=\csc x\cot x+3x$

18) $f(x)=4\csc x\cot x−\sec x\tan x$

Answer

$F(x)=−\sec x−4\csc x+C$

19) $f(x)=8(\sec x)\big(\sec x−4\tan x\big)$

20) $f(x)=\frac{1}{2}e^{−4x}+\sin x$

Answer

$F(x)=−\frac{1}{8}e^{−4x}−\cos x+C$

For exercises 21 - 29, evaluate the integral.

21) $\displaystyle ∫(−1)\,dx$

22) $\displaystyle ∫\sin x\,dx$

Answer

$\displaystyle ∫\sin x\,dx = −\cos x+C$

23) $\displaystyle ∫\big(4x+\sqrt{x}\big)\,dx$

24) $\displaystyle ∫\frac{3x^2+2}{x^2}\,dx$

Answer

$\displaystyle ∫\frac{3x^2+2}{x^2}\,dx=3x−\frac{2}{x}+C$

25) $\displaystyle ∫\big(\sec x\tan x+4x\big)\,dx$

26) $\displaystyle ∫\big(4\sqrt{x}+\sqrt[4]{x}\big)\,dx$

Answer

$\displaystyle ∫\big(4\sqrt{x}+\sqrt[4]{x}\big)\,dx=\frac{8}{3}x^{3/2}+\frac{4}{5}x^{5/4}+C$

27) $\displaystyle ∫\left(x^{−1/3}−x^{2/3}\right)\,dx$

28) $\displaystyle ∫\frac{14x^3+2x+1}{x^3}\,dx$

Answer

$\displaystyle ∫\frac{14x^3+2x+1}{x^3}\,dx=14x−\frac{2}{x}−\frac{1}{2x^2}+C$

29) $\displaystyle ∫\big(e^x+e^{−x}\big)\,dx$

In exercises 30 - 34, solve the initial value problem.

30) $f′(x)=x^{−3},\quad f(1)=1$

Answer

$f(x)=−\dfrac{1}{2x^2}+\dfrac{3}{2}$

31) $f′(x)=\sqrt{x}+x^2,\quad f(0)=2$

32) $f′(x)=\cos x+\sec^2(x),\quad f(\frac{π}{4})=2+\frac{\sqrt{2}}{2}$

Answer

$f(x)=\sin x+\tan x+1$

33) $f′(x)=x^3−8x^2+16x+1,\quad f(0)=0$

34) $f′(x)=\dfrac{2}{x^2}−\dfrac{x^2}{2},\quad f(1)=0$

Answer

$f(x)=−\frac{1}{6}x^3−\dfrac{2}{x}+\dfrac{13}{6}$

In exercises 35 - 39, find two possible functions $f$ given the second- or third-order derivatives

35) $f''(x)=x^2+2$

36) $f''(x)=e^{−x}$

Answer

Answers may vary; one possible answer is $f(x)=e^{−x}$

37) $f''(x)=1+x$

38) $f'''(x)=\cos x$

Answer

Answers may vary; one possible answer is $f(x)=−\sin x$

39) $f'''(x)=8e^{−2x}−\sin x$

40) A car is being driven at a rate of $40$ mph when the brakes are applied. The car decelerates at a constant rate of $10\, \text{ft/sec}^2$. How long before the car stops?

Answer

$5.867$ sec

41) In the preceding problem, calculate how far the car travels in the time it takes to stop.

42) You are merging onto the freeway, accelerating at a constant rate of $12\, \text{ft/sec}^2$. How long does it take you to reach merging speed at $60$ mph?

Answer

$7.333$ sec

43) Based on the previous problem, how far does the car travel to reach merging speed?

44) A car company wants to ensure its newest model can stop in $8$ sec when traveling at $75$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

Answer

$13.75\, \text{ft/sec}^2$

45) A car company wants to ensure its newest model can stop in less than $450$ ft when traveling at $60$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

In exercises 46 - 51, find the antiderivative of the function, assuming $F(0)=0.$

46) [T] $\quad f(x)=x^2+2$

Answer

$F(x)=\frac{1}{3}x^3+2x$

47) [T] $\quad f(x)=4x−\sqrt{x}$

48) [T] $\quad f(x)=\sin x+2x$

Answer

$F(x)=x^2−\cos x+1$

49) [T] $\quad f(x)=e^x$

50) [T] $\quad f(x)=\dfrac{1}{(x+1)^2}$

Answer

$F(x)=−\dfrac{1}{x+1}+1$

51) [T] $\quad f(x)=e^{−2x}+3x^2$

In exercises 52 - 55, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.

52) If $f(x)$ is the antiderivative of $v(x)$, then $2f(x)$ is the antiderivative of $2v(x).$

Answer

True

53) If $f(x)$ is the antiderivative of $v(x)$, then $f(2x)$ is the antiderivative of $v(2x).$

54) If $f(x)$ is the antiderivative of $v(x),$ then $f(x)+1$ is the antiderivative of $v(x)+1.$

Answer

False

55) If $f(x)$ is the antiderivative of $v(x)$, then $(f(x))^2$ is the antiderivative of $(v(x))^2.$