Definite Integration
Integral of cos^a(x)cos(bx) — reduction
MJAT_TS4_P1
Grade 12
Question:
For non-negative integers $a$ and $b$, let $I(a,b)=\displaystyle\int_0^{\pi/2}\cos^a x\cos bx\,dx$.
Match each entry in List-I to the correct entry in List-II.
**List-I:** P) $I(0,5)$; Q) $I(1,4)$; R) $I(2,4)$; S) $I(3,2)$
**List-II:** 1) $\dfrac{1}{5}$; 2) $\dfrac{1}{5}I(2,1)$; 3) $\dfrac{1}{4}[I(1,3)-I(1,5)]$; 4) $\dfrac{1}{5}I(0,3)$; 5) $\dfrac{1}{3}I(2,1)$
A) P→1, Q→4, R→3, S→2
B) P→3, Q→1, R→5, S→2
C) P→2, Q→1, R→5, S→3
D) P→1, Q→4, R→1, S→3
Step-by-Step Solution
Key Concept: Use the reduction formula: $I(a,b)=\frac{a}{a+b}I(a-1,b-1)+\frac{a}{a-b}I(a-1,b+1)$ (Wallis-type). Or use $I(a,b)=\frac{a}{2(a+b)}I(a-1,b-1)$ directly.
Answer: **A**. P→(1), Q→(4), R→(3), S→(2).
Correct Answer: A