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# 4.8E: AntiDerivative & Indefinite Integral Exercises

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## 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) $$f(x)=e^x−3x^2+sinx$$

$$F(x)=e^x−x^3−cos(x)+C$$

472) $$f(x)=e^x+3x−x^2$$

473) $$f(x)=x−1+4sin(2x)$$

$$F(x)=\frac{x^2}{2}−x−2cos(2x)+C$$

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

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

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

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

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

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

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

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

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

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

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

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

482) $$f(x)=sinxcosx$$

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

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

484) $$f(x)=0$$

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

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

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

487) $$f(x)=4cscxcotx−secxtanx$$

$$F(x)=−secx−4cscx+C$$

488) $$f(x)=8secx(secx−4tanx)$$

489) $$f(x)=\frac{1}{2}e^{−4x}+sinx$$

$$F(x)=−\frac{1}{8}e^{−4x}−cosx+C$$

For the following exercises, evaluate the integral.

490) $$∫(−1)dx$$

491) $$∫sinxdx$$

$$−cosx+C$$

492) $$∫(4x+\sqrt{x})dx$$

493) $$∫\frac{3x^2+2}{x^2}dx$$

$$3x−\frac{2}{x}+C$$

494) $$∫(secxtanx+4x)dx$$

495) $$∫(4\sqrt{x}+\sqrt{x})dx$$

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

496) $$∫(x^{−1/3}−x^{2/3})dx$$

497) $$∫\frac{14x^3+2x+1}{x^3}dx$$

$$14x−\frac{2}{x}−\frac{1}{2x^2}+C$$

498) $$∫(e^x+e^{−x})dx$$

For the following exercises, solve the initial value problem.

499) $$f′(x)=x^{−3},f(1)=1$$

$$f(x)=−\frac{1}{2x^2}+\frac{3}{2}$$

500) $$f′(x)=\sqrt{x}+x^2,f(0)=2$$

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

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

502) $$f′(x)=x^3−8x^2+16x+1,f(0)=0$$

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

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

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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 $$(v(x))^2.$$