← 2021 Paper 1

UPSC 2021 Maths Optional Paper 1 Q3a-iii — Step-by-Step Solution

8 marks · Section A

Indefinite integrals · Calculus · asked 7× in 14 yrs · Read the full method →

Question

Express $\int_a^b(x-a)^m(b-x)^n\,dx$ in terms of the Beta function.

Technique

Substitute $x=a+(b-a)t$ to map $[a,b]\to[0,1]$; the factor $(b-a)^{m+n+1}$ comes from $(x-a)^m,(b-x)^n,dx$.

Solution

Beta function definition: $B(p,q)=\int_0^1 t^{p-1}(1-t)^{q-1}\,dt$, $p,q>0$.

Step 1 — Substitute $x=a+(b-a)t$, $t\in[0,1]$

$dx=(b-a)\,dt$.

$x-a=(b-a)t$, so $(x-a)^m=(b-a)^m t^m$.

$b-x=(b-a)(1-t)$, so $(b-x)^n=(b-a)^n(1-t)^n$.

$\int_a^b(x-a)^m(b-x)^n\,dx=\int_0^1(b-a)^m t^m\cdot(b-a)^n(1-t)^n\cdot(b-a)\,dt$

$=(b-a)^{m+n+1}\int_0^1 t^m(1-t)^n\,dt$

$=(b-a)^{m+n+1}\cdot B(m+1,n+1)$.

Answer