Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

4.8E: AntiDerivative & Indefinite Integral Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

4.8: Antiderivatives

For the following exercises, find the antiderivative F(x) of each function f(x).

470) f(x)=1x2+x

471) f(x)=ex3x2+sinx

Answer:
F(x)=exx3cos(x)+C

472) f(x)=ex+3xx2

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

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

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

475) f(x)=x+12x2

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

476) f(x)=1x

477) f(x)=(x)3

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

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

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

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

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

481) f(x)=sec2(x)+1

Answer:
(F(x)=x+tan(x)+C\)

482) f(x)=sinxcosx

483) f(x)=sin2(x)cos(x)

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

484) f(x)=0

485) f(x)=12csc2(x)+1x2

Answer:
F(x)=12cot(x)1x+C

486) f(x)=cscxcotx+3x

487) f(x)=4cscxcotxsecxtanx

Answer:
F(x)=secx4cscx+C

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

489) f(x)=12e4x+sinx

Answer:
F(x)=18e4xcosx+C

For the following exercises, evaluate the integral.

490) (1)dx

491) sinxdx

Answer:
cosx+C

492) (4x+x)dx

493) 3x2+2x2dx

Answer:
3x2x+C

494) (secxtanx+4x)dx

495) (4x+4x)dx

Answer:
83x3/2+45x5/4+C

496) (x1/3x2/3)dx

497) 14x3+2x+1x3dx

Answer:
14x2x12x2+C

498) (ex+ex)dx

For the following exercises, solve the initial value problem.

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

Answer:
f(x)=12x2+32

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

501) f(x)=cosx+sec2(x),f(π4)=2+22

Answer:
f(x)=sinx+tanx+1

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

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

Answer:
f(x)=16x32x+136

J4.8.1) not here yet

Answer:
8

J4.8.2) not here yet

J4.8.3) not here yet

Answer:
8

For the following exercises, find two possible functions f given the second- or third-order derivatives

504) f

505) f''(x)=e^{−x}

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

506) f''(x)=1+x

507) f'''(x)=cosx

Solution: Answers may vary; one possible answer is f(x)=−sinx

508) f'''(x)=8e^{−2x}−sinx

509) 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 ft/sec^2. How long before the car stops?

Solution: 5.867 sec

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

511) You are merging onto the freeway, accelerating at a constant rate of 12 ft/sec2. How long does it take you to reach merging speed at 60 mph?

Solution: 7.333 sec

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

513) 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.

Solution: 13.75 ft/sec^2

514) 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.

For the following exercises, find the antiderivative of the function, assuming F(0)=0.

515) [T] f(x)=x^2+2

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

516) [T] f(x)=4x−\sqrt{x}

517) [T] f(x)=sinx+2x

Solution: F(x)=x^2−cosx+1

518) [T] f(x)=e^x

519) [T] f(x)=\frac{1}{(x+1)^2}

Solution: F(x)=−\frac{1}{(x+1)}+1

520) [T] f(x)=e^{−2x}+3x^2

For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.

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

Solution: True

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

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

Solution: False

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


4.8E: AntiDerivative & Indefinite Integral Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

Support Center

How can we help?