Prove $\int_0^a (a^2 - x^2)^n dx = a^{2n+1}\cdot\frac{(2n)!!}{(2n+1)!!}$

Question

Prove $$\int_0^a (a^2 - x^2)^n dx = a^{2n+1}\cdot\frac{(2n)!!}{(2n+1)!!}$$ for $ n \in N $ My idea is to use binomial theorem and integrate partly, but I have not succeeded yet. Are there some hints?

Answer 1

Note: @Heropup's approach seems to be the most adequate. But you can also succeed by applying the binomial theorem.

We obtain \begin{align*} \int_{0}^a\left(a^2-x^2\right)^n\,dx&=\int_0^a\sum_{k=0}^n\binom{n}{k}\left(-x^2\right)^k\left(a^2\right)^{n-k}\,dx\\ &=\sum_{k=0}^n\binom{n}{k}(-1)^ka^{2n-2k}\int_0^ax^{2k}\,dx\\ &=\sum_{k=0}^n\binom{n}{k}(-1)^ka^{2n-2k}\frac{1}{2k+1}a^{2k+1}\\ &=a^{2n+1}\sum_{k=0}^n(-1)^k\binom{n}{k}\frac{1}{2k+1}\tag{1} \end{align*}

There is a nice identity called Melzak's formula which can be used to solve (1).

Melzak's formula: Let $f(x)$ be a polynomial in $x$ of degree $n$, i.e. \begin{align*} f(x)=\sum_{k=0}^na_kx^k \end{align*} Let $y$ be an arbitrary complex number. Melzak's formula states that for $y \ne 0,-1,-2,\ldots,-n$ \begin{align*} f(x+y)&=y\binom{y+n}{n}\sum_{k=0}^n(-1)^k\binom{n}{k}\frac{f(x-k)}{y+k}\tag{2}\\ \end{align*}

The whole chapter 7 in H.W.Goulds Combinatorial Identities for Stirling Numbers is devoted to this formula. An easily accessible proof can be found in New proofs of Melzak's identity by U. Abel, H.W. Gould and J. Quaintance.

Setting $f(x)=1$ and $y=\frac{1}{2}$ we obtain from (2) \begin{align*} \sum_{k=0}^n(-1)^k\binom{n}{k}\frac{1}{2k+1}=\frac{1}{\binom{\frac{1}{2}+n}{n}} \end{align*}

Since \begin{align*} \binom{\frac{1}{2}+n}{n}&=\frac{\left(n+\frac{1}{2}\right)\left(n-\frac{1}{2}\right)\cdots\left(n+\frac{1}{2}-(n-1)\right)}{n!}\\ &=\frac{1}{2^n}\cdot\frac{\left(2n+1\right)\left(2n-1\right)\cdots\left(3\right)}{n!}\\ &=\frac{(2n+1)!!}{2^nn!}\\ &=\frac{(2n+1)!!}{(2n)!!} \end{align*} the claim follows.

Answer 2

First hint: consider the case where $a = 1$.

Second hint: Let $$I_n = \int_{x=0}^1 (1 - x^2)^n \, dx = \int_{x=0}^1 (1-x^2)^{n-1} - x^2 (1-x^2)^{n-1} \, dx = I_{n-1} - \int_{x=0}^1 x (x(1-x^2)^{n-1}) \, dx,$$ then with this last integral, perform integration by parts with the choice $$u = x, \quad du = dx, \quad dv = x(1-x^2)^{n-1} \, dx, \quad v = -\frac{1}{2n}(1-x^2)^n.$$ This yields a recursion relation for $I_n$, which you can then solve.

Answer 3

Take out $a$ as a common factor, giving $\frac{1}{a^{2n+1}} \, \int_0^1 (1 - y^2)^n dy = a^{2n+1} \int_0^1 (1 - y^2)^n dy$ then use the integral over $y$ is standard (or easily determined) and given as $\frac{(2n)!!}{(2n+1)!!}$.