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  4. If f(x)=∫ (5 x8+7 x6/(x2+1+2 x7)2) d x, and f(0)=0, then the value of f(1) is

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

Integrals

Solution:

$f(x) =\int \frac{5 x^{8}+7 x^{6}}{x^{14}\left(2+\frac{1}{x^{7}}+\frac{1}{x^{5}}\right)^{2}} d x$
$=\int \frac{\frac{5}{x^{6}}+\frac{7}{x^{8}}}{\left(2+\frac{1}{x^{7}}+\frac{1}{x^{5}}\right)^{2}} d x$
Put, $2+\frac{1}{x^{7}}+\frac{1}{x^{5}}=t$
$\Rightarrow f(x)=-\int \frac{d t}{t^{2}}$
$=\frac{1}{t}+c$
$=\left(\frac{x^{7}}{2 x^{7}+x^{2}+1}\right)+c$
$f(0)=0$
$\Rightarrow c=0$
$\Rightarrow f(x)=\left(\frac{x^{7}}{2 x^{7}+x^{2}+1}\right)$
$\Rightarrow f(1)=\frac{1}{4}$