Question

$\rm \Large{Hello all !}$

$\boxed{\red{\rm \int \dfrac{x^9}{(4x^2+1)^6}\ dx}}$

Solve the above integral , Thank you !

Answer 1

$\displaystyle \sf \red{ \int \dfrac{x^9}{(4x^2+1)^6}\ dx}$

Taking x² common in the denominator ,

$\longrightarrow \displaystyle \sf \int \dfrac{x^9}{\left[ x^2\left( 4+ \frac{1}{x^2}\right) \right]^6}\ dx$

$\longrightarrow \displaystyle \sf \int \dfrac{x^9}{x^{12}\left[ \left( 4+ \frac{1}{x^2}\right) \right]^6}\ dx$

$\longrightarrow \displaystyle \sf \int \dfrac{1}{x^{3}\left[ \left( 4+ \frac{1}{x^2}\right) \right]^6}\ dx$

Using Substitution method ,

Let  $\sf u=\left( 4+ \dfrac{1}{x^2} \right)$

$\implies \sf - \dfrac{du}{2} = \dfrac{1}{x^3}\ dx$

$\longrightarrow \displaystyle \sf \int \dfrac{1}{u^6} \left( - \dfrac{du}{2} \right)$

$\longrightarrow \displaystyle \sf - \dfrac{1}{2}\int \dfrac{1}{u^6}\ du$

$\longrightarrow \displaystyle \sf - \dfrac{1}{2} \left[ \dfrac{u^{-5}}{-5} \right]\ +c$

$\longrightarrow \displaystyle \sf \dfrac{1}{10u^5}\ +c$

$\longrightarrow \sf \orange{\dfrac{1}{10\ \left( 4+ \dfrac{1}{x^2} \right)^5}\ +c}\ \; \bigstar$