JEE Main 2016 — Mathematics Calculus

Given, $\displaystyle I=\int \frac{2{x}^{12}+5{x}^{9}}{{({x}^{5}+{x}^{3}+1)}^{3}}dx$I=∫(x5+x3+1)32x12+5x9​dx

$$=\int \frac{(2{x}^{12}+5{x}^{9})dx}{{x}^{15}{(1+\frac{1}{{x}^{2}}+\frac{1}{{x}^{5}})}^{3}}$$=∫x15(1+x21​+x51​)3(2x12+5x9)dx​

$$=\int \frac{(\frac{2}{{x}^{3}}+\frac{5}{{x}^{6}})dx}{{(1+\frac{1}{{x}^{2}}+\frac{1}{{x}^{5}})}^{3}}$$=∫(1+x21​+x51​)3(x32​+x65​)dx​

Let, $1+\frac{1}{{x}^{2}}+\frac{1}{{x}^{5}}=t$1+x21​+x51​=t $\Rightarrow (-\frac{2}{{x}^{3}}-\frac{5}{{x}^{6}})dx=dt$⇒(−x32​−x65​)dx=dt

$$\Rightarrow (\frac{2}{{x}^{3}}+\frac{5}{{x}^{6}})dx=-dt$$⇒(x32​+x65​)dx=−dt

$$I=-\int \frac{dt}{{t}^{3}}=\frac{1}{2 . {t}^{2}}$$I=−∫t3dt​=2.t21​

$$I=\frac{1}{2}\frac{1}{{(1+\frac{1}{{x}^{2}}+\frac{1}{{x}^{5}})}^{2}}+c=\frac{1}{2}\frac{{x}^{10}}{{({x}^{5}+{x}^{3}+1)}^{2}}+c$$I=21​(1+x21​+x51​)21​+c=21​(x5+x3+1)2x10​+c

Where $c$c is the constant of integration.

$$=\frac{{x}^{10}}{2{({x}^{5}+{x}^{3}+1)}^{2}}+c$$=2(x5+x3+1)2x10​+c