Integral Calculus - Exercises

1 Integral Calculus - Indefinite IntegralIn problems 1 through 7,find the indicated xdx=Zx12dx=23x32+C=23x x+ + (3x2 5x+2) (3x2 5x+2)dx=3Zx2dx 5Z xdx+2 Zdx==3 13x3 5 23x x+2x+C==x3 23x 5x+2x+ 12x 2x2+3 x 12x 2x2+3 x dx=12Z1xdx 2Zx 2dx+3Zx 12dx==12ln|x| 2 ( 1)x 1+3 2x12+C==ln|x|2+2x+6 x+ Calculus - 2ex+6x+ln2 2ex+6x+ln2 dx=2 Zexdx+6Z1xdx+ln2 Zdx==2ex+6ln|x|+(ln2)x+ +3x 2 +3x 2 xdx=Zx32dx+3Zx12dx 2Zx 12dx==25x52+3 23x32 2 2x12+C==25x52+2x32 4x12+C==25x2 x+2x x 4 x+ (x3 2x2) 1x 5 (x3 2x2) 1x 5 dx=Z(x2 5x3 2x+10x2)dx==Z( 5x3+11x2 2x)dx== 5 14x4+11 13x3 2 12x2+C== 54x4+113x3 x2+ Find the functionfwhose tangent has slopex3 2x2+2for each valueofxand whose graph passes through the point(1,3). slope of the tangent is the derivative (x)=x3 2x2+2and sof(x)is the indefinite integralf(x)=Zf0(x)dx=Z x3 2x2+2 dx==14x4+2x+2x+ Calculus - EXERCISES42 Using the fact that the graph offpasses through the point(1,3)youget3=14+2+2+CorC= , the desired function isf(x)=14x4+2x+2x It is estimated thattyears from now the population of a certain lakesidecommunity will be changing at the rate + + per year.

2 Environmentalists have found that the level of pollu-tion in the lake increases at the rate of approximately5units per1000people. By how much will the pollution in the lake increase during thenext2years? (t)denote the population of the communitytyearsfrom now. Then the rate of change of the population with respect totime is the derivativedPdt=P0(t)= + + follows that the population functionP(t)is an antiderivative + + ,P(t)=ZP0(t)dt=Z( + + )dt== + + +Cfor some constantC. During the next 2 years, the population will growon behalf ofP(2) P(0) = 23+ 22+ 2+C C== + +1=3thousand , the pollution in the lake will increase on behalf of5 3= Anobjectismovingsothatitsspeedaftertminu tes isv(t)=1+4t+3t2meters per minute. How far does the object travel during3rd minute? (t)denote the displacement of the car (t)=dsdt=s0(t)it follows thats(t)=Zv(t)dt=Z(1 + 4t+3t2)dt=t+2t2+t3+ the 3rd minute, the object travelss(3) s(2) = 3 + 2 9+27+C 2 2 4 8 C== Calculus - EXERCISES43 HomeworkIn problems 1 through 13,find the indicated Integral .

3 Check your answersby (x12 3x23+6) 3 x 2x3+1x ex2+x x x3 12 x+ 2 13x 32x2+e2+ x2 +2x+ 2x+1x x(x2 1) (2x+1)2dx14. Find the function whose tangent has slope4x+1for each value ofxand whose graph passes through the point(1,2).15. Find the function whose tangent has slope3x2+6x 2for each valueofxand whose graph passes through the point(0,6).16. Find a function whose graph has a relative minimum whenx=1anda relative maximum whenx= It is estimated thattmonths from now the population of a certain townwill be changing at the rate of4+5t23people per month. If the currentpopulation is 10000, what will the population be8months from now?18. An environmental study of a certain community suggests thattyearsfrom now the level of carbon monoxide in the air will be changing + per million per year. If the current level ofcarbon monoxide in the air per million, what will the levelbe3years from now?

4 19. After its brakes are applied, a certain car decelerates at the constantrate of6meters per second per second. If the car is traveling at108kilometers per hour when the brakes are applied, how far does it travelbefore coming to a complete stop? (Note:108kmph is the same as30mps.)20. Suppose a certain car supplies a constant deceleration ofAmeters persecond per second. If it is traveling at90kilometersperhour(25meters per second) when the brakes are applied, its stopping distanceis50meters.(a) What isA? Integral Calculus - EXERCISES44(b) What would the stopping distance have been if the car had beentraveling at only54kilometers per hour when the brakes wereapplied?(c) At what speed is the car traveling when the brakes are applied ifthe stopping distance is56meters? + +C3. 1x+ + 95x53+6x+ +1x2+ln|x|+ +25x52+ (x3)x x+ 2x+ |x|+32x+e2x+13x32+ 1x+2lnx+ +13x3+ 23x32+ +43x3+12x2+ (x)=2x2+x (x)=x3+3x2 2x+ (x)=13x3 52x2+4x.

5 Not per (a)A= (b)42meters(c) per hourINTEGRAL Calculus - Integration by SubstitutionIn problems 1 through 8,find the indicated (2x+6) +6and12du=dx,yougetZ(2x+6)5dx=12Zu5du=11 2u6+C=112(2x+6)6+ [(x 1)5+3(x 1)2+5] 1anddu=dx,yougetZ (x 1)5+3(x 1)2+5 dx=Z(u5+3u2+5)du==16u6+u3+5u+C==16(x 1)6+(x 1)3+5(x 1) + , for a constantC,C 5is again a constant, you can writeZ (x 1)5+3(x 1)2+5 dx=16(x 1)6+(x 1)3+5x+ ,yougetZxex2dx=12 Zeudu=12eu+C=12ex2+ x6and 16du=x5dx,yougetZx5e1 x6dx= 16 Zeudu= 16eu+C= 16e1 x6+ + +1and25du=2x4dx, you getZ2x4x5+1dx=25Z1udu=25ln|u|+C=25ln x5+1 + Calculus - 5x x4 x2+ x2+6and52du=(10x3 5x)dx,yougetZ10x3 5x x4 x2+6dx=52Z1 udu=52Zu 12du=52 2u12+C==5 x4 x2+6+ ,yougetZ1xlnxdx=Z1udu=ln|u|+C=ln|lnx|+ ,yougetZlnx2xdx=Z2lnxxdx=2 Zudu=2 12u2+C=(lnx)2+ Use an appropriate change of variables tofind the integralZ(x+1)(x 2) 2,u+3=x+1anddu=dx, you getZ(x+1)(x 2)9dx=Z(u+3)u9du=Z(u10+3u9)du==111u11+31 0u10+C==111(x 2)11+310(x 2)10+ Use an appropriate change of variables tofind the integralZ(2x+3) 2x 1,u+4=2x+3and12du=dx,youINTEGRAL Calculus - EXERCISES47getZ(2x+3) 2x 1dx=12Z(u+4) udu=12Zu32du+2Zu12du==12 25u52+2 23u32+C=15u52+43u32+C==15(2x 1)52+43(2x 1)32+C==15(2x 1)2 2x 1+43(2x 1) 2x 1+C==(2x 1) 2x 1 25x 15+43 +C== 25x+1725 (2x 1) 2x 1+ problems 1 through 18,find the indicated Integral and check your answerby 4x + (x2+1) x2+ (x3+1) (x3+5) (x+1)(x2+2x+5) (3x2 1)ex3 +12x3+6x5+5x4+10x+ 3(x2 2x+6) 34x2 4x+ (lnx) (x2+1)x2+ x xdxIn problems 19 through 23, use an appropriate change of variables tofindthe indicated x+ (x 5) +3(x 4) +1dx24.

6 Find the function whose tangent has slopex x2+5for each value ofxand whose graph passes through the point(2,10).25. Find the function whose tangent has slope2x1 3x2for each value ofxandwhose graph passes through the point(0,5). Integral Calculus - EXERCISES4826. A tree has been transplanted and afterxyears is growing at the rateof1+1(x+1)2meters per year. After two years it has reached a heightoffive meters. How tall was it when it was transplanted?27. It is projected thattyears from now the population of a certain countrywill be changing at the rate per year. If the currentpopulation is50million, what will the population be10years fromnow? + (4x 1) 4x 1+ |3x+5|+C4. e1 x+ 1+ (x2+1)6+C7.(x2+8) x2+8+ (x3+1)74+C9. 13(x3+5)+ (x2+2x+5)13+ x+ |x5+5x4+10x+12|+C13. 32(x2 2x+6)+ |2x 1|+ +C16. 1lnx+ (x2+1)+ x+ +ln|x 1|+ (x+1)2 x+1 23(x+1) x+1+C21. 1(x 5)5 14(x 5)4+C22. 7x 4+ln|x 4|+ +14x 14ln|2x+1|+ (x)=13(x2+5) x2+5+ (x)= 13ln|1 3x2|+ Calculus - Integration by PartsIn problems 1 through 9, use integration by parts tofind the given the easy to integrate and the factorxissimplified by differentiation, try integration by parts withg(x)= (x)= ,G(x)= (x)=1and +C== 10(x 10) + (3 2x)e the factore xis easy to integrate and the factor3 2xis simplified by differentiation, try integration by parts withg(x)=e xandf(x)=3 ,G(x)=Ze xdx= e xandf0(x)= 2and soZ(3 2x)e xdx=(3 2x)( e x) 2Ze xdx==(2x 3)e x+2e x+C=(2x 1)e x+ this case, the factorxis easy to integrate, while thefactorlnx2is simplified by differentiation.

7 This suggests that you tryintegration by parts withg(x)=xandf(x)= ,G(x)=Zxdx=12x2andf0(x)=1x22x=2xINTEGRAL Calculus - EXERCISES50and soZxlnx2dx=12x2lnx2 Z12x22xdx=12x2lnx2 Zxdx==12x2lnx2 12x2+C=12x2 lnx2 1 + 1 the factor 1 xis easy to integrate and the factorxis simplified by differentiation, try integration by parts withg(x)= 1 xandf(x)= ,G(x)=Z 1 xdx= 23(1 x)32andf0(x)=1and soZx 1 xdx= 23x(1 x)32+23Z(1 x)32dx== 23x(1 x)32+23 25(1 x)52 +C== 23x(1 x)32 415(1 x)52+C== 23x(1 x) 1 x 415(1 x)2 1 x+ (x+1)(x+2) the factor(x+2)6is easy to integrate and the factorx+1is simplified by differentiation, try integration by parts withg(x)=(x+2)6andf(x)=x+ ,G(x)=Z(x+2)6dx=17(x+2)7andf0(x)=1and soZ(x+1)(x+2)6dx=17(x+1)(x+2)7 17Z(x+2)7dx==17(x+1)(x+2)7 1718(x+2)8+C==156[8(x+1) (x+2)](x+2)7+C==156(7x+6)(x+2)7+ Calculus - the factore2xis easy to integrate and the factorx3issimplified by differentiation, try integration by parts withg(x)=e2xandf(x)= ,G(x)=Ze2xdx=12e2xandf0(x)=3x2and soZx3e2xdx=12x3e2x , you have to integrate by parts again, but this timewithg(x)=e2xandf(x)= ,G(x)=12e2xandf0(x)=2xand soZx2e2xdx=12x2e2x , you have to integrate by parts once again, this timewithg(x)=e2xandf(x)= ,G(x)=12e2xandf0(x)=1and soZxe2xdx=12xe2x 12Ze2xdx=12xe2x ,Zx3e2xdx=12x3e2x 32 12x2e2x 12xe2x 14e2x +C== 12x3 34x2+34x 38 e2x+ Calculus - this case, the factor1x3is easy to integrate, while thefactorlnxis simplified by differentiation.

8 This suggests that you tryintegration by parts withg(x)=1x3andf(x)= ,G(x)=Z1x3dx= 12x 2= 12x2andf0(x)=1xand soZlnxx3dx= lnx2x2+12Z1x3dx= lnx2x2+12 12x2 +C== lnx2x2 14x2+ rewrite the integrand asx2 xex2 , and then integrateby parts withg(x)=xex2andf(x)= , from Exercise you getG(x)=Zxex2dx=12ex2andf0(x)=2xand soZx3ex2dx=12x2ex2 Zxex2=12x2ex2 12ex2+C==12(x2 1)ex2+ (x2 1) rewrite the integrand asx2[x(x2 1)10],andthenintegrate by parts withg(x)=x(x2 1)10andf(x)= (x)=Zx(x2 1)10dxandf0(x)= Calculus - EXERCISES53 Substitutingu=x2 1and12du=xdx,yougetG(x)=Zx(x2 1)10dx=12Zu10du=122u11=122(x2 1) (x2 1)10dx=122x2(x2 1)11 122Z2x(x2 1)11dx==122x2(x2 1)11 122112(x2 1)12+C==122x2(x2 1)11 1264(x2 1)12+C.(a) Use integration by parts to derive the formulaZxneaxdx=1axneax naZxn 1eaxdx.(b) Use the formula in part (a) (a) Since the factoreaxis easy to integrate and thefactorxnis simplified by differentiation, try integration by partswithg(x)=eaxandf(x)= ,G(x)=Zeaxdx=1aeaxandf0(x)=nxn 1and soZxneaxdx=1axneax naZxn 1eaxdx.

9 (b) Apply the formula in part (a) witha=5andn=3to getZx3e5xdx=15x3e5x , apply the formula in part (a) witha=5andn=2tofindthe new integralZx2e5xdx=15x2e5x Calculus - EXERCISES54 Once again, apply the formula in part (a) witha=5andn=1to getZxe5xdx=15xe5x 15Ze5xdx=15xe5x 125e5xand soZx3e5xdx=15x3e5x 35 15x2e5x 25 15xe5x 125e5x +C==15 x3 35x2+625x 6125 e5x+ problems 1 through 16, use integration by parts tofind the given (1 x) x (x+1) x+ 2x+ (lnx) (x4+5)8dx17. Find the function whose tangent has slope(x+1)e xfor each value ofxand whose graph passes through the point(1,5).18. Find the function whose tangent has slopexln xfor each value ofx>0and whose graph passes through the point(2, 3).19. Aftertseconds, an object is moving at the speed ofte t2meters persecond. Express the distance the object travels as a function of It is projected thattyears from now the population of a certain citywill be changing at the rate oftln t+1thousand people per year.

10 Ifthe current population is2million, what will the population be5yearsfrom now? Integral Calculus - (x+1)e x+C2.(2x 4)e12x+C3. 5(x+5)e 15x+C4.(2 x)ex+ (ln 2x 12)+ (x 6)32 415(x 6)52+ (x+1)9 190(x+1)10+ (x+2)12 43(x+2)32+ (2x+1)12 13(2x+1)32+C10. (x2+2x+2)e x+ x2 23x+29 e3x+C12.(x3 3x2+6x 6)ex+ 19x3+ ln2x lnx+12 +C15. 1x(lnx+1)+ (x4+5)9 1360(x4+5)10+ (x)= (x+2)e x+3e+ (x)=14x2 lnx 12 52 ln (t)= 2(t+2)e t2+ Calculus - The use of Integral tablesIn Problems 1 through 5, use one of the integration formulas from a table ofintegrals (see Appendix) tofind the given x2+2x rewrite the integrand as1 x2+2x 3=1p(x+1)2 4andthensubstituteu=x+1anddu=dxto getZdx x2+2x 3=Zdxp(x+1)2 4=Zdu u2 4==ln u+ u2 4 +C=ln x+1+p(x+1)2 4 +C==ln x+1+ x2+2x 3 + 6x , rewrite the integrand as11 6x 3x2=11 3(2x+x2)=14 3(x+1)2=13143 (x+1)2andthensubstituteu=x+1anddu=dxto getZdx1 6x 3x2=13 Zdx43 (x+1)2=13 Zdu43 u2==1338ln 43+u43 u +C=18ln 4+3u4 3u +C==18ln 7+3x1 3x + (x2+1) rewrite the integrand as(x2+1)32=(x2+1) x2+1=x2 x2+1+ x2+ appropriate formulas (see Appendix, formulas 9 and 13), to getZx2 x2+1dx=x8(1 + 2x2)