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  4. If m is a non - zero number and ∫ (x5m-1 + 2x4m - 1/(x2m +xm +1)3)dx = f(x) +c, then f(x) is :

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Q. If m is a non - zero number and $\int \frac{x^{5m-1} + 2x^{4m - 1}}{\left(x^{2m }+x^{m} +1\right)^{3}}dx = f\left(x\right) +c,$
then f(x) is :

JEE MainJEE Main 2014Integrals

Solution:

$\int \frac{x^{5m-1} + 2x^{4m - 1}}{\left(x^{2m }+x^{m} +1\right)^{3}}dx$
$= \int \frac{x^{5m-1} + 2x^{4m - 1}}{x^{6m}\left(1+x^{-m }+x^{-2m}\right)^{3}}dx$
$= \int \frac{x^{-m-1} + 2x^{-2m - 1}}{\left(1+x^{-m }+x^{-2m}\right)^{3}}dx$
Put $t = 1+x^{-m }+x^{-2m}$
$\therefore \frac{dt}{dx} = -mx^{-m-1}-2mx^{-2m-1}$
$\Rightarrow \frac{dt}{-m} = \left(x^{-m-1}+2x^{-2m-1}\right)dx$
$\therefore \int \frac{x^{5m-1} + 2x^{4m - 1}}{\left(x^{2m }+x^{m} +1\right)^{3}}dx$
$= \frac{1}{-m}\int t^{-3} dt = \frac{1}{2mt^{2}}+C$
$= \frac{1}{2m\left(1+x^{-m}+x^{-2m}\right)}+C$
$= \frac{x^{4m}}{2m\left(x^{2m}+x^{m}+1\right)^{2}}+C$
$\therefore f\left(x\right) = \frac{x^{4m}}{2m\left(x^{2m}+x^{m}+1\right)^{2}}$