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

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Aicharya · Accepted Answer

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

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…

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