My dear Beta-Gamma

Calculus Level 5

$\large \int _0^3 {x^\frac 12} \left(27- x^3 \right)^{-\frac 12} dx$

The definite integral above has a closed form of $\dfrac \pi n$ . Find $n$ .

9 3 2 13 8 15 7 6

2 solutions

Kushal Bose
Nov 26, 2016

Put $x^3=27z$ .Then it will become known form of Beta Function.

First Last
Nov 25, 2016

A u-subsitution $\displaystyle x^3=u,\quad du = 3x^2dx \quad$ so the integral becomes

$\displaystyle\frac{1}{3}\int_{0}^{27}u^\frac{-1}{2}(27-u)^\frac{-1}{2}du = \frac{\pi}{3}\quad$ by noting that

$\displaystyle\int_{0}^{a}\frac{1}{\sqrt{x}\sqrt{a-x}} = \arcsin{( \frac{2x}{a}-1)}|^a_0 = \pi$