The integral∫2x12+5x9(x5+x3+1)3 dx is equal to :

I2x3+5x61+1x2+1x53dx,Â Â Â letÂ 1+1x2+1x5=t HenceÂ I=x102x5+x3+12+c

JEE Main | 2016Math miscellaneousHard

Question

The integral $\int \frac{2 x^{12} + 5 x^{9}}{(x^{5} + x^{3} + 1)^{3}}$ dx is equal to :

Options

$\frac{x^{10}}{2 (x^{5} + x^{3} + 1)^{2}} + c$

$\frac{x^{5}}{2 (x^{5} + x^{3} + 1)^{2}} + c$

$\frac{- x^{10}}{2 (x^{5} + x^{3} + 1)^{2}} = c$

$\frac{- x^{5}}{(x^{5} + x^{3} + 1)^{2}} = c$

Solution

$I \{\begin{cases} \frac{(\begin{matrix} \frac{2}{x^{3}} + \frac{5}{x^{6}} \end{matrix})}{{(\begin{matrix} 1 + \frac{1}{x^{2}} + \frac{1}{x^{5}} \end{matrix})}^{3}} d x, Â Â Â l e t Â 1 + \frac{1}{x^{2}} + \frac{1}{x^{5}} = t \end{cases}$
$H e n c e Â I = \frac{x^{10}}{2 {(\begin{matrix} x^{5} + x^{3} + 1 \end{matrix})}^{2}} + c$

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