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Let $d,n~$ be positive integer.

$\int{y(1-y^d)^n}dy$

Is it possible to solve it? If you know the method, please teach me.

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    Perhaps a binomial integral is what you're looking for http://www.nabla.hr/CL-IndefIntegralB5.htm – 2017-01-17

2 Answers 2

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Hint. Expand the term $(1-y^d)^n$ using the binomial theorem, and multiply by $y$. Then you will simply have a polynomial, and polynomials are easy to integrate.

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As Eff answered, the binomial theorem seems to be the way to do it.

You will learn sooner or later that $$\int{y(1-y^d)^n}\,dy=\frac{1}{2} y^2 \,\, _2F_1\left(\frac{2}{d},-n;\frac{d+2}{d};y^d\right)$$ where appears the Gaussian or ordinary hypergeometric function (see here).