Integrate ((x / (1+x²)) ?

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Answer 1 · 2018-03-17T00:51:31

#frac(1)(2) ln|1 + x^(2)| + C#

We have: #int# #frac(x)(1 + x^(2))# #dx#

Let #u = 1 + x^(2) Rightarrow du = 2x# #dx#:

#= frac(1)(2) int# #frac(x)(1 + x^(2)) cdot 2# #dx#

#= frac(1)(2) int# #frac(1)(1 + x^(2)) cdot 2x# #dx#

#= frac(1)(2) int# #frac(1)(u) cdot# #du#

#= frac(1)(2) cdot ln|u| + C#

We can now replace #u# with #1 + x^(2)#:

#= frac(1)(2) ln|1 + x^(2)| + C#