If \(f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+1+2 x^7\right)^2} d x, x \geq 0\) and \(f(0)=0\), then value of \(f(1)\) is

\(\begin{aligned} & f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+1+2 x^7\right)^2} d x \\ & =\int \frac{5 x^8+7 x^6}{x^{14}\left(\frac{1}{x^5}+\frac{1}{x^7}+2\right)^2} d x \\ & =\int \frac{5 x^{-6}+7 x^{-8}}{\left(\frac{1}{x^5}+\frac{1}{x^7}+2\right)^2} d x \\ & =-\int \frac{d t}{t^2}\left[\text { where } t=\frac{1}{x^5}+\frac{1}{x^7}+2\right] \\ & =\frac{1}{t}+C\end{aligned}\)
\(\begin{aligned} & =\frac{1}{\left(\frac{1}{x^5}+\frac{1}{x^7}+2\right)}+C \\ & \Rightarrow f(x)=\frac{x^7}{x^2+1+2 x^7}+C \\ & \because f(0)=0 \Rightarrow c=0 \text { i.e. } f(x)=\frac{x^7}{x^2+1+2 x^7} \\ & \Rightarrow f(1)=\frac{1}{4}\end{aligned}\)