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).

2 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. Environmentalists have found that the level of pollu-tion in the lake increases at the rate of approximately5units per1000people.

3 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.

4 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 . 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).

5 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.

6 If the current level ofcarbon monoxide in the air per million, what will the levelbe3years from now?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.

7 (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; 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)

8 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)

9 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. 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.

10 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)