$$ \int \frac { x ^ { 2 } d x } { ( x ^ { 2 } + 1 ) ^ { r } } $$

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Perkins Conner · Accepted Answer

The integral $$ \int \frac{x^2}{(x^2 + 1)^r} \, dx $$ can be solved using substitution. Let $$ u = x^2 + 1 $$, then $$ du = 2x \, dx $$. Substituting gives: $$ \frac{1}{2} \int \frac{u - 1}{u^r \sqrt{u - 1}} \, du $$ This can be split into two integrals: $$ \frac{1}{2} \int \frac{u}{u^r \sqrt{u - 1}} \, du - \frac{1}{2} \int \frac{1}{u^r \sqrt{u - 1}} \, du $$ Depending on the value of $$ r $$, further integration techniques may be needed.

\( \int \frac { x ^ { 2 } d x } { ( x ^ { 2 } + 1 ) ^ { r } } \)

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\( \int \frac { x ^ { 2 } d x } { ( x ^ { 2 } + 1 ) ^ { r } } \)

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