Integral Calculus
Integration by parts; evaluating P(π/3)
MMTS_Full_Test_05
Grade 12
Question:
If $\displaystyle\int\frac{\sec^2 x-2010}{\sin^{2010}x}\,dx=\frac{P(x)}{(\sin x)^{2010}}+C$, then the value of $P\!\left(\dfrac{\pi}{3}\right)$ is
(A) 0
(B) $\dfrac{1}{\sqrt{3}}$
(C) $\sqrt{3}$
(D) $\dfrac{3\sqrt{3}}{2}$
Step-by-Step Solution
Key Concept: Split: $\int\sec^2x/\sin^{2010}x\,dx - 2010\int 1/\sin^{2010}x\,dx$. Apply IBP on $I_1$: $\int\sec^2x/\sin^{2010}x\,dx=\tan x/\sin^{2010}x+2010\int\tan x\cos x/\sin^{2011}x\,dx$.
$P(x)=\tan x$, $P(\pi/3)=\sqrt{3}$.
Correct Answer: (C) $\sqrt{3}$