4.8E: AntiDerivative

4.8: Antiderivatives

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

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

471) $f(x)=e^x−3x^2+sinx$

Answer:

$F(x)=e^x−x^3−cos(x)+C$

472) $f(x)=e^x+3x−x^2$

473) $f(x)=x−1+4sin(2x)$

Answer:

$F(x)=\frac{x^2}{2}−x−2cos(2x)+C$

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

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

Answer:

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

476) $f(x)=\frac{1}{\sqrt{x}}$

477) $f(x)=(\sqrt{x})^3$

Answer:

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

478) $f(x)=x^{1/3}+(2x)^{1/3}$

479) $f(x)=\frac{x^{1/3}}{x^{2/3}}$

Answer:

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

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

481) $f(x)=sec^2(x)+1$

Answer:

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

482) $f(x)=sinxcosx$

483) $f(x)=sin^2(x)cos(x)$

Answer:

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

484) $f(x)=0$

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

Answer:

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

486) $f(x)=cscxcotx+3x$

487) $f(x)=4cscxcotx−secxtanx$

Answer:

$F(x)=−secx−4cscx+C$

488) $f(x)=8secx(secx−4tanx)$

489) $f(x)=\frac{1}{2}e^{−4x}+sinx$

Answer:

$F(x)=−\frac{1}{8}e^{−4x}−cosx+C$

For the following exercises, evaluate the integral.

490) $∫(−1)dx$

491) $∫sinxdx$

Answer:

$−cosx+C$

492) $∫(4x+\sqrt{x})dx$

493) $∫\frac{3x^2+2}{x^2}dx$

Answer:

$3x−\frac{2}{x}+C$

494) $∫(secxtanx+4x)dx$

495) $∫(4\sqrt{x}+\sqrt[4]{x})dx$

Answer:

$\frac{8}{3}x^{3/2}+\frac{4}{5}x^{5/4}+C$

496) $∫(x^{−1/3}−x^{2/3})dx$

497) $∫\frac{14x^3+2x+1}{x^3}dx$

Answer:

$14x−\frac{2}{x}−\frac{1}{2x^2}+C$

498) $∫(e^x+e^{−x})dx$

For the following exercises, solve the initial value problem.

499) $f′(x)=x^{−3},f(1)=1$

Answer:

$f(x)=−\frac{1}{2x^2}+\frac{3}{2}$

500) $f′(x)=\sqrt{x}+x^2,f(0)=2$

501) $f′(x)=cosx+sec^2(x),f(\frac{π}{4})=2+\frac{\sqrt{2}}{2}$

Answer:

$f(x)=sinx+tanx+1$

502) $f′(x)=x^3−8x^2+16x+1,f(0)=0$

503) $f′(x)=\frac{2}{x^2}−\frac{x^2}{2},f(1)=0$

Answer:

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

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

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 $(v(x))^2.$