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Review of integration by substitution: Use u-substitution to find each of the following integrals.

a. e3x+1dxb. xex2+1dx

c. lnxxdxd. 1xlnxdx

e. tanxsec2xdxf. 1x2+1dx

g.sinxcosxdxh.exsinexdx

Short Answer

Expert verified

Part (a)e3x+1dx=e3x+13+C

Part (b)xex2+1dx=ex2+12+C

Part (c)lnxxdx=lnx22+C

Part (d)1xlnxdx=lnlnx+C

Part (e)tanxsec2xdx=tan2x2+C

Part (f)1x2+1dx=tan-1x+C

Part (g)sinxcosxdx=-2cos32x3+C

Part (h)exsinexdx=-cosex+C

Step by step solution

01

Part (a) Step 1. Calculating the integral 

Given integral, e3x+1dx

Substituting, 3x+1=u,3dx=du

eu3du=eu3+C=e3x+13+C

02

Part (b) Step 1. Calculating the integral 

Given integral, xex2+1dx

Substituting, x2+1=u,2xdx=du

eu2du=eu2+C=ex2+12+C

03

Part (c) Step 1. Calculating the integral 

Given integral, lnxxdx

Substituting, lnx=u,dxx=du

udu=u22+C=lnx22+C

04

Part (d) Step 1. Calculating the integral 

Given integral, 1xlnxdx

Substituting, lnx=u,dxx=du

1udu=lnu+C=lnlnx+C

05

Part (e) Step 1. Calculating the integral 

Given integral, tanxsec2xdx

Substituting, tanx=u,sec2xdx=du

udu=u22+C=tan2x2+C

06

Part (f) Step 1. Calculating the integral 

Given integral, 1x2+1dx

=tan-1x+C

07

Part (g) Step 1. Calculating the integral 

Given integral, sinxcosxdx

Substituting, cosx=u,-sinxdx=du

-udu=-u3232+C=-2cos32x3+C

08

Part (h) Step 1. Calculating the integral 

Given integral, exsinexdx

Substituting, ex=u,exdx=du

sinudu=-cosu+C=-cosex+C

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