Problem 3: Evaluate $\displaystyle \int \dfrac{x^3 \, dx}{(x^2 + 1)^3}$ by Algebraic Substitution

$\displaystyle \int \dfrac{x^3 \, dx}{(x^2 + 1)^3}$

$\qquad = {\displaystyle \int}\dfrac{x^2 \, (x \, dx)}{(x^2 + 1)^3}$
 

Let
$u^2 = x^2 + 1$

$x^2 = u^2 - 1$

$2x\,dx = 2u\,du$

$x\,dx = u\,du$

 

Hence,
$\displaystyle \int \dfrac{x^3 \, dx}{(x^2 + 1)^3}$

$\qquad = {\displaystyle \int}\dfrac{(u^2 - 1) \, (u\,du)}{(u^2)^3}$

$\qquad = {\displaystyle \int}\dfrac{(u^3 - u) \, du}{u^6}$

$\qquad = {\displaystyle \int}\left( \dfrac{1}{u^3} - \dfrac{1}{u^5} \right) \, du$

$\qquad = {\displaystyle \int}(u^{-3} - u^{-5}) \, du$

$\qquad = \dfrac{u^{-2}}{-2} - \dfrac{u^{-4}}{-4} + C$

$\qquad = -\dfrac{1}{2u^2} + \dfrac{1}{4u^4} + C$
 

Revert to the original variable of integration
$\displaystyle \int \dfrac{x^3 \, dx}{(x^2 + 1)^3}$

$\qquad = -\dfrac{1}{2(x^2 + 1)} + \dfrac{1}{4(x^2 + 1)^2} + C$   ←   answer