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  4. If x f(x)=3(f(x))2+2, then ∫ (2 x2-12 x f(x)+f(x)/(6 f(x)-x)(x2-f(x))2) d x is equal to

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Q. If $x f(x)=3(f(x))^2+2$, then $\int \frac{2 x^2-12 x f(x)+f(x)}{(6 f(x)-x)\left(x^2-f(x)\right)^2} d x$ is equal to

Integrals

Solution:

$f(x)+x f^{\prime}(x)=6 f(x) \cdot f^{\prime}(x) $
$\Rightarrow f(x)=(6 f(x)-x) f^{\prime}(x) $
$ \text { Now I }=\int \frac{2 x(x-6 f(x))+f(x)}{(6 f(x)-x)\left(x^2-f(x)\right)^2} d x$
$-\int \frac{2 x(6 f(x)-x)-(6 f(x)-x) \cdot f^{\prime}(x)}{(6 f(x)-x)\left(x^2-f(x)\right)^2} d x$
$=-\int \frac{2 x-f^{\prime}(x)}{\left(x^2-f(x)\right)^2} d x$
put $x^2-f(x)=t$
$\left(2 x-f^{\prime}(x) d x=d t\right.$
so $I=\frac{1}{x^2-f(x)}+C$