Practice Integration Z Math 120 Calculus I

1 Practice IntegrationMath 120 Calculus ID Joyce, Fall 2013 This first set of indefinite integrals, that is, an-tiderivatives, only depends on a few principles ofintegration, the first being that Integration is in-verse to differentiation. Besides that, a few rulescan be identified: a constant rule, a power rule,linearity, and a limited few rules for trigonometric,logarithmic, and exponential functions. k dx=kx+C,wherekis a constant xndx=1n+ 1xn+1+C,ifn6= 1 1xdx= ln|x|+C kf(x)dx=k f(x)dx (f(x) g(x))dx= f(x)dx g(x)dx sinxdx= cosx+C cosxdx= sinx+C exdx=ex+C 11 +x2dx= arctanx+C 1 1 x2dx= arcsinx+CWe ll add more rules later, but there are plenty hereto get acquainted s a list of Practice exercises. There s a hintfor each one as well as an answer with (x4 x3+x2)dx.

2 Hint. (5t8 2t4+t+ 3)dt. Hint. (7u3/2+ 2u1/2)du. Hint. (3x 2 4x 3)dx. Hint. 3xdx. Hint. (43t2+72t)dt. Hint. (5 y 3 y)dy. Hint. 3x2+ 4x+ 12xdx. Hint. (2 sin + 3 cos )d . Hint. (5ex e)dx. Hint. 41 +t2dt. Hint. (ex+3+ex 3)dx. Hint. 7 1 u2du. Hint. (r2 2r+1r)dr. Hint. 4 sinx3 tanxdx. Hint. (7 cosx+ 4ex)dx. Hint. 3 7v dv. Hint. 4 5tdt. Hint. 13x2+ 3dx. Hint. x4 6x3+ex x xdx. Hint. Hint. (x4 x3+x2) each term using the power rule, xndx=1n+ 1xn+1+ to integratexn, increase the power by 1, thendivide by the new power. Hint. (5t8 2t4+t+ 3) that the integral of a constant is theconstant times the integral. Another way to saythat is that you can pass a constant through theintegral sign.

3 For instance, 5t8dt= 5 t8dtIntegrating polynomials is fairly easy, and you llget the hang of it after doing just a couple of Hint. (7u3/2+ 2u1/2) can use the power rule for other powers be-sides integers. For instance, u3/2du=25u5/2+ Hint. (3x 2 4x 3)dxYou can even use the power rule for negative ex-ponents (except 1). For example, x 3dx= 12x 2+ Hint. 3xdxThis is 3x 1and the general power rule doesn tapply. But you can use 1xdx= ln|x|+ Hint. (43t2+72t)dtTreat the first term as43t 2and the second termas72t 1. Hint. (5 y 3 y)dyIt s usually easier to turn those square roots intofractional powers. So, for instance,1 yisy 1 Hint. 3x2+ 4x+ 12xdxUse some algebra to simplify the integrand, thatis, divide by 2xbefore integrating. Hint. (2 sin + 3 cos )d Getting the signs right when integrating sinesand cosines takes Practice .

4 Hint. (5ex e)dxJust as the derivative ofexisex, so the integralofexisex. Note that the ein the integrand is aconstant. Hint. 41 +t2dtRemember that the derivative of arctantis11 +t2. Hint. (ex+3+ex 3)dxWhen working with exponential functions, re-member to use the various rules of exponentia-tion. Here, the rules to use areea+b=eaebandea b=ea/eb. Hint. 7 1 u2duRemember that the derivative of arcsinuis1 1 Hint. (r2 2r+1r)drUse the power rule, but don t forget the integralof 1/ris ln|r|+C. Hint. 4 sinx3 tanxdxYou ll need to use trig identities to simplify Hint. (7 cosx+ 4ex)dxJust more Practice with trig and exponentialfunctions. Hint. 3 7v dvYou can write3 7vas3 73 v. And rememberyou can write3 vasv1/3. Hint.

5 4 5tdtUse algebra to write this in a form that s easier tointegrate. Remember that 1/ tist 1/2. Hint. 13x2+ 3dxYou can factor out a 3 from the denominator toput it in a form you can integrate. Hint. x4 6x3+ex x xdxDivide through by xbefore integrating. Alter-natively, write the integrand asx 1/2(x4 6x3+exx1/2)and multiply. Answer. (x4 x3+x2) integral is15x5 14x4+13x3+ you re working with indefinite inte-grals like this, be sure to write the +C. It signifiesthat you can add any constant to the antiderivativeF(x) to get another one,F(x) + you re working with definite integrals withlimits of Integration , ba, the constant isn t neededsince you ll be evaluating an antiderivativeF(x) atbandato get a numerical answerF(b) F(a).32. Answer. (5t8 2t4+t+ 3) integral is59t9 25t5+12t2+ 3t+ Answer.

6 (7u3/2+ 2u1/2) integral evaluates as145u5/2+43u3/2+ Answer. (3x 2 4x 3) equals 3x 1+ 2x 2+C. If you prefer, youcould write the answer as 3x+2x2+C5. Answer. 3xdxThat s 3 ln|x|+C. The reason the absolute valuesign is there is that whenxis negative, the deriva-tive of ln|x|is 1/x, so by putting in the absolutevalue sign, you re covering that case, Answer. (43t2+72t) integral of43t 2+72t 1is 43t 1+72ln|t|+ Answer. (5 y 3 y) integral of 5y1/2 3y 1/2is103y3/2 6y1/2+ could write that as103y y 6 y+Cif Answer. 3x2+ 4x+ integral of 2x+ 2 +12x 1isx2+ 2x+12ln|x|+ Answer. (2 sin + 3 cos )d .That s equal to 2 cos + 3 sin + Answer. (5ex e)dxThat equals 5ex ex+ Answer. 41 + evaluates as 4 arctant+C. Some peopleprefer to write arctantas tan Answer. (ex+3+ex 3) integrand is its own antiderivative, that is,the integral is equal toex+3+ex 3+ you write the integrand asexe3+ex/e3, and notethate3is just a constant, you can see that it s itsown Answer.

7 7 1 integral equals 7 Answer. (r2 2r+1r) integral evaluates as13r3 r2+ ln|r|+ Answer. 4 sinx3 tanxdxThe integrand simplifies to43cosx. Therefore theintegral is43sinx+ Answer. (7 cosx+ 4ex) s 7 sinx+ 4ex+ Answer. 3 7v you can rewrite the integrand as3 7v1/3,therefore its integral is343 7v4/3+ Answer. 4 integral of4 5t 1/2is equal to8 5t1/2+ could also write that as 8 t/5 + Answer. 13x2+ 3dxThis integral equals13arctanx+ Answer. x4 6x3+ex x integral can be rewritten as (x7/2 6x5/2+ex)dxwhich equals29x9/2 127x7/2+ex+ 120 Home Page ~djoyce/ma120/5