Definite Integration
Integration by Parts Recursive Formula
nta_pyq_2023_apr
Grade 12
Question:
For $m,n>0$, let $\alpha(m,n)=\displaystyle\int_0^2 t^m(1+3t)^n\,dt$. If $11\alpha(10,6)+18\alpha(11,5)=p(14)^6$, then $p$ is equal to
Step-by-Step Solution
Key Concept: Apply integration by parts to $\alpha(m,n)$: $(m+1)\alpha(m,n)+3n\alpha(m+1,n-1)=7^n\cdot 2^{m+1}$. This is a reduction formula.
Reduction: $11\alpha(10,6)+18\alpha(11,5)=7^6\cdot 2^{11}=32\cdot14^6$. $p=32$.
Correct Answer: 32