Beta function problems

Question

Calculate the integral:

$$I = \int_0^1 (1-x^m)^{q-1} \cdot x^{p-1} \, dx$$

Please help me!I now the shapes of beta function but I can't solve this exercises.

Answer 1

I'm assuming you want to write $$I = \int_0^1 (1-x^m)^{q-1} \cdot x^{p-1} \, dx$$ in terms of a Beta function.

---

Let $u = x^m$. Then $$du = mx^{m-1} \; dx = mu^\frac{m-1}{m} \; dx$$ Substituting into $I$, \begin{align*} I &= \int_0^1 (1-u)^{q-1} \cdot u^\frac{p-1}{m} \cdot \frac{1}{m} u^\frac{1-m}{m} \; du \\ &= \frac{1}{m} \int_0^1 (1-u)^{q-1} u^{\frac{p}{m}-1} \; du \\ &= \frac{1}{m} B\left(q, \frac{p}{m}\right) \end{align*}