Question

Integrate x^15/(1+x^32). What is the solution for this?

Answer 1

We are given to find,

$\displaystyle\longrightarrow\sf{\int\dfrac{x^{15}}{x^{32}+1}\ dx}$

$\displaystyle\longrightarrow\sf{\dfrac{1}{16}\int\dfrac{16x^{15}}{(x^{16})^2+1}\ dx\quad\quad\dots(1)}$

Let,

$\displaystyle\longrightarrow\sf{u=x^{16}}$

$\displaystyle\longrightarrow\sf{\dfrac{du}{dx}=16x^{15}}$

$\displaystyle\longrightarrow\sf{dx=\dfrac{1}{16x^{15}}\ du}$

Hence (1) becomes,

$\displaystyle\longrightarrow\sf{\dfrac{1}{16}\int\dfrac{16x^{15}}{u^2+1}\cdot\dfrac{1}{16x^{15}}\ du}$

$\displaystyle\longrightarrow\sf{\dfrac{1}{16}\int\dfrac{1}{u^2+1}\ du}$

$\displaystyle\longrightarrow\sf{\dfrac{\tan^{-1}(u)}{16}+c}$

$\displaystyle\longrightarrow\sf{\underline{\underline{\dfrac{tan^{-1}(x^{16})}{16}+c}}}$