Question:
If $\quad f(x)=\int \frac{5 x^{8}+7 x^{6}}{\left(x^{2}+1+2 x^{7}\right)^{2}} d x,(x \geq 0)$
and
$f(0)=0$, then the value of $f(1)$ is :
Correct Option: , 4
Solution:
$\int \frac{5 x^{8}+7 x^{6}}{\left(x^{2}+1+2 x^{7}\right)^{2}} d x$
$=\int \frac{5 x^{-6}+7 x^{-8}}{\left(\frac{1}{x^{7}}+\frac{1}{x^{5}}+2\right)^{2}} d x=\frac{1}{2+\frac{1}{x^{5}}+\frac{1}{x^{7}}}+C$
As $f(0)=0, f(x)=\frac{x^{7}}{2 x^{7}+x^{2}+1}$
$f(1)=\frac{1}{4}$