Originally Posted by
Mrhappysmile I'm just not sure how to go about solving this:
In= ∫(x^n).(1-x)^(3/2)dx
The integral has limits from 1 -> 0
And I need to show :
In = (2n/2n+5) I(n-1)
Integrate by parts, using the fact that $\displaystyle \int(1-x)^{3/2} = -\tfrac25(1-x)^{5/2}$. Then
$\displaystyle \begin{aligned}I_n &= \Bigl[-\tfrac25(1-x)^{5/2}x^n\Bigr]_0^1 + \int_0^1nx^{n-1}\tfrac25(1-x)^{5/2}dx \\ &= \tfrac{2n}5\int_0^1 x^{n-1}(1-x)(1-x)^{3/2}dx = \tfrac{2n}5I_{n-1} - \tfrac{2n}5I_n. \end{aligned}$