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If $f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+2 x^7+1\right)^2} d x(x \geq 0)$ and $f(0)=0$, then the value of $f(1)=$
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Verified Answer
The correct answer is:
$\frac{1}{4}$
$\begin{aligned}
& \text {Given } f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+2 x^7+1\right)^2} d x \\
& \Rightarrow f(x)=\int \frac{5 x^{-6}+7 x^{-8}}{\left(\frac{1}{x^7}+\frac{1}{x^5}+2\right)^2}=\frac{1}{2+\frac{1}{x^7}+\frac{1}{x^5}}+c
\end{aligned}$
Since,
$f(0)=0 \Rightarrow c=0 \Rightarrow f(1)=\frac{1}{4}$
& \text {Given } f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+2 x^7+1\right)^2} d x \\
& \Rightarrow f(x)=\int \frac{5 x^{-6}+7 x^{-8}}{\left(\frac{1}{x^7}+\frac{1}{x^5}+2\right)^2}=\frac{1}{2+\frac{1}{x^7}+\frac{1}{x^5}}+c
\end{aligned}$
Since,
$f(0)=0 \Rightarrow c=0 \Rightarrow f(1)=\frac{1}{4}$
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