Integral 2x^3dx/(x^2 +1)^2 =ln(x^2 +1)+1/x^2 +1 +C? Thanksyou

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Answer 1 · 2018-01-29T08:17:53

#ln(x^2+1)+1/(x^2+1)+C#

We want to solve the integral

#int(2x^3)/(x^2+1)^2dx#

Use substitution, let #u=x^2+1# then #(du)/dx=2x#

#int(2x^3)/u^2*1/(2x)du#

Simplify

#int(x^2)/u^2du#

Use #u=x^2+1<=>x^2=u-1#

#int(u-1)/(u^2)du#

Use the sum rule for integrals

#int1/udu-int1/u^2du#

Do the integration

#ln(u)+1/u+C#

Substitute #u=x^2+1#

#ln(x^2+1)+1/(x^2+1)+C#