how can i solve the following integral

Question

$$I = \int _{a}^{a+h} x^n \alpha h^{-\alpha}(x-a)^{\alpha-1} dx$$

$$I = \alpha h^{-\alpha} \int_{a}^{a+h}x^n (x-a)^{\alpha-1} dx$$

The result will be

$$\sum_{k=0}^{n} {n \choose k} h^k a^{n-k} \frac{\alpha}{\alpha+k}$$

Answer 1

Make the substitution $x=a+y$, $dx=dy$. The integral becomes

$$\alpha h^{-\alpha} \int_0^h dy \, (a+y)^n y^{\alpha-1}$$

You can expand the binomial, assuming $n \in \mathbb{N}$:

$$(a+y)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}y^k$$

You then reverse order of sum and integration and get

$$\alpha h^{-\alpha} \sum_{k=0}^n \binom{n}{k} a^{n-k} \int_0^h dy \, y^{k+\alpha-1} = \alpha h^{-\alpha} \sum_{k=0}^n \binom{n}{k} a^{n-k} \frac{h^{k + \alpha}}{k+\alpha}$$

The result follows.