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Integration: Calculate the following integrals.

a.1x-12dxb.x-1x2-1dx

c.1x2+1dxd.xx2+1dx

e.x2+1x3dxf.x+1x2+1dx

Short Answer

Expert verified

Part (a)1x-12dx=-1x-1+C

Part (b)x-1x2-1dx=ln|x+1|+C

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

Part (d)xx2+1dx=ln|x2+1|2+C

Part (e)x2+1x3dx=lnx-12x2+C

Part (f)x+1x2+1dx=lnx2+12+tan-1x+C

Step by step solution

01

Part (a) Step 1. Calculating the integral

Given integral,1x-12dx

=x-1-2+1(-2+1)=-1x-1+C

02

Part (b) Step 1. Calculating the integral

Given integral, x-1x2-1dx

=x-1x-1x+1dx=1x+1dx=ln|x+1|+C

03

Part (c) Step 1. Calculating the integral

Given integral, 1x2+1dx

=tan-1x+C

04

Part (d) Step 1. Calculating the integral

Given integral, xx2+1dx

=2x2x2+1dx=ln|x2+1|2+C

05

Part (e) Step 1. Calculating the integral

Given integral, x2+1x3dx

=x2x3+1x3dx=1x+1x3dx=lnx+x-3+1(-3+1)+C=lnx-12x2+C

06

Part (f) Step 1. Calculating the integral

Given integral, x+1x2+1dx

=xx2+1+1x2+1dx=2x2x2+1+1x2+1dx=lnx2+12+tan-1x+C

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