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4.11E: Antiderivative and Indefinite Integral Exercises

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

1) f(x)=1x2+x

2) f(x)=ex3x2+sinx

Answer
F(x)=exx3cosx+C

3) f(x)=ex+3xx2

4) f(x)=x1+4sin(2x)

Answer
F(x)=x22x2cos(2x)+C

5) f(x)=5x4+4x5

6) f(x)=x+12x2

Answer
F(x)=12x2+4x3+C

7) f(x)=1x

8) f(x)=(x)3

Answer
F(x)=25(x)5+C

9) f(x)=x1/3+(2x)1/3

10) f(x)=x1/3x2/3

Answer
F(x)=32x2/3+C

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

12) f(x)=sec2x+1

Answer
F(x)=x+tanx+C

13) f(x)=sinxcosx

14) f(x)=sin2(x)cos(x)

Answer
F(x)=13sin3(x)+C

15) f(x)=0

16) f(x)=12csc2x+1x2

Answer
F(x)=12cotx1x+C

17) f(x)=cscxcotx+3x

18) f(x)=4cscxcotxsecxtanx

Answer
F(x)=secx4cscx+C

19) f(x)=8(secx)(secx4tanx)

20) f(x)=12e4x+sinx

Answer
F(x)=18e4xcosx+C

For exercises 21 - 29, evaluate the integral.

21) (1)dx

22) sinxdx

Answer
sinxdx=cosx+C

23) (4x+x)dx

24) 3x2+2x2dx

Answer
3x2+2x2dx=3x2x+C

25) (secxtanx+4x)dx

26) (4x+4x)dx

Answer
(4x+4x)dx=83x3/2+45x5/4+C

27) (x1/3x2/3)dx

28) 14x3+2x+1x3dx

Answer
14x3+2x+1x3dx=14x2x12x2+C

29) (ex+ex)dx

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

30) f(x)=x3,f(1)=1

Answer
f(x)=12x2+32

31) f(x)=x+x2,f(0)=2

32) f(x)=cosx+sec2(x),f(π4)=2+22

Answer
f(x)=sinx+tanx+1

33) f(x)=x38x2+16x+1,f(0)=0

34) f(x)=2x2x22,f(1)=0

Answer
f(x)=16x32x+136

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

35) f

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.

 


4.11E: Antiderivative and Indefinite Integral Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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