Adela Brown

Answered

2022-01-05

Evaluate $\int \frac{1}{{({x}^{2}+1)}^{2}}dx$

Answer & Explanation

nghodlokl

Expert

2022-01-06Added 33 answers

Set

Laura Worden

Expert

2022-01-07Added 45 answers

There is a faster way. Substitute

$x=\mathrm{tan}\left(z\right)\text{}dx={\mathrm{sec}}^{2}\left(z\right)dz$

thus

${({x}^{2}+1)}^{2}={({\mathrm{tan}}^{2}\left(z\right)+1)}^{2}={\mathrm{sec}}^{4}\left(z\right)\text{}z=\mathrm{arctan}\left(x\right)$

And remembering that

$\frac{1}{{\mathrm{sec}}^{2}}={\mathrm{cos}}^{2}$

your integral is simply

$\int {\mathrm{cos}}^{2}\left(z\right)dz$

Which is trivial and left to you.

The tangent/secant substitution is a great technique which most of people ignore. Study it, and you will solve lots of awesome integrals!

Final result:

$\frac{{x}^{2}\mathrm{arctan}\left(x\right)+x+\mathrm{arctan}\left(x\right)}{2{x}^{2}+2}$

thus

And remembering that

your integral is simply

Which is trivial and left to you.

The tangent/secant substitution is a great technique which most of people ignore. Study it, and you will solve lots of awesome integrals!

Final result:

star233

Expert

2022-01-11Added 238 answers

Let us find

Integrating both sides,

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