if f(x) =


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 the value of $f(1)$ is:

  1. (1) $-\frac{1}{2}$

  2. (2) $-\frac{1}{4}$

  3. (3) $\frac{1}{2}$

  4. (4) $\frac{1}{4}$

Correct Option: , 4



$=\int \frac{5 x^{8}+7 x^{6}}{x^{14}\left(x^{-5}+x^{-7}+2\right)^{2}} d x$

$=\int \frac{5 x^{-6}+7 x^{-8}}{\left(2+x^{-7}+x^{-5}\right)^{2}} d x$

Let $2+x^{-7}+x^{-5}=t$

$\Rightarrow \quad\left(-7 x^{-8}-5 x^{-6}\right) d x=d t$

$\Rightarrow f(x)=\int \frac{-d t}{t^{2}}=\int-t^{-2} d t=t^{-1}+c$

$\Rightarrow f(x)=\frac{1}{2+x^{-7}+x^{-5}}+c, f(0)=0 \Rightarrow c=0$

$\therefore \quad f(1)=\frac{1}{4}$

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