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  4. If ∫ (1/x+x5)dx =f(x)+c , then ∫ (x4/x+x5) dx =

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Q. If $\int \frac{1}{x+x^{5}}dx =f\left(x\right)+c , $ then $\int \frac{x^{4}}{x+x^{5}} dx = $

COMEDKCOMEDK 2008Integrals

Solution:

$\int \frac{1}{x+x^{5}}dx =f\left(x\right)+c $
$ I = \int \frac{x^{4}}{x+x^{5}} dx = \int \frac{x^{4} +1 -1}{x \left(1+x^{4}\right)} dx$
$ = \int \frac{x^4+1}{x(1+x^4)} dx - \int \frac{1}{x(1+x^4)} dx$
$ = \int \frac{1}{x} dx - \int \frac{1}{x+x^5} dx $
$= log |x| - f(x) + c$