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  4. If ∫ (dx/x+x7) = p(x) then, ∫ (x6/x+x7)dx is equal to:

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Q. If $\int \frac{dx}{x+x^{7}} = p\left(x\right)$ then, $\int \frac{x^{6}}{x+x^{7}}dx$ is equal to:

JEE MainJEE Main 2013Integrals

Solution:

$\int \frac{x^{6}}{x+x^{7}}dx = \int \frac{x^{6}}{x\left(1+x^{6}\right)}dx$
$= \int \frac{\left(1+x^{6}\right)-1}{x\left(1+x^{6}\right)}dx$
$=\int \frac{1}{x}dx-\int \frac{1}{x+x^{7}}dx$
$= In \left|x\right| -p\left(x\right) + c$