Definite Integration
Beta Function
nta_pyq_2024_apr
Grade 12
Question:
Let $\beta(m,n)=\int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m,n>0$. If $\int_0^1(1-x^{10})^{20}\,dx=a\cdot\beta(b,c)$, then $100(a+b+c)$ equals:
Step-by-Step Solution
Key Concept: Substitute $x^{10}=t$: $dx=\frac{1}{10}t^{-9/10}dt$. Integral $=\frac{1}{10}\int_0^1 t^{1/10-1}(1-t)^{21-1}dt=\frac{1}{10}\beta(1/10,21)$.
$a=1/10$, $b=1/10$, $c=21$. $100(a+b+c)=2120$.
Correct Answer: 2