Definite Integration
Complex trig integral — constants identification
MJAT_TS6_P2
Grade 12
Question:
If $\displaystyle\int_{\pi/4}^{\pi/3}\frac{(\sin^3x-\cos^3x-\cos^2x)(\sin^{2027}x+\cos^{2027}x)^{2025}}{\sin^{2025}x\cos^{2025}x}\,dx=(a+b)^{2026}-\!\left(1+d+\frac{1}{c}\right)^{\!2026}$, $a,b,c,d\in\mathbb{N}$. Which is/are correct?
A) $a+b=5$
B) $c=6$
C) $d$ is odd
D) Number of positive integral divisors of $d$ is $15$
Step-by-Step Solution
Key Concept: Let $u=\tan x+\cot x$... or substitute $t=\sin x/\cos x=\tan x$. The integrand simplifies to a perfect power. Result: $a=2,b=3,c=8,d=2025$.
A ✓, C ✓, D ✓. Answer: A, C, D.
Correct Answer: ACD