5.3E: Exercises for Section 5.3

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

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

2) $$f(x)=e^x−3x^2+\sin x$$

$$F(x)=e^x−x^3−\cos x+C$$

3) $$f(x)=e^x+3x−x^2$$

4) $$f(x)=x−1+4\sin(2x)$$

$$F(x)=\dfrac{x^2}{2}−x−2\cos(2x)+C$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

15) $$f(x)=0$$

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

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

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

18) $$f(x)=4\csc x\cot x−\sec x\tan x$$

$$F(x)=−\sec x−4\csc x+C$$

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

20) $$f(x)=\frac{1}{2}e^{−4x}+\sin x$$

$$F(x)=−\frac{1}{8}e^{−4x}−\cos x+C$$

For exercises 21 - 29, evaluate the integral.

21) $$\displaystyle ∫(−1)\,dx$$

22) $$\displaystyle ∫\sin x\,dx$$

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

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

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

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

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

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

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

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

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

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

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

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

30) $$f′(x)=x^{−3},\quad f(1)=1$$

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

31) $$f′(x)=\sqrt{x}+x^2,\quad f(0)=2$$

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

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

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

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

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

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

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

36) $$f''(x)=e^{−x}$$

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

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

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

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?

$$5.867$$ 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?

$$7.333$$ 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.

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

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

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

47) [T] $$\quad f(x)=4x−\sqrt{x}$$

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

$$F(x)=x^2−\cos x+1$$

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

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

$$F(x)=−\dfrac{1}{x+1}+1$$

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 $$2v(x).$$

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 $$v(x)+1.$$

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