Definite Integration
Function Defined by Integral
nta_pyq_2024_apr
Grade 12
Question:
If $f(t)=\int_0^\pi\dfrac{2x\,dx}{1-\cos^2t\sin^2x}$, $0<t<\pi$, then the value of $\int_0^{\pi/2}\dfrac{\pi^2\,dt}{f(t)}$ equals ________.
Step-by-Step Solution
Key Concept: Use King's property on $f(t)$: add $f(t)$ with substitution $x\to\pi-x$ to get $f(t)=\pi\int_0^\pi\frac{dx}{1-\cos^2t\sin^2x}$. After simplification using $\tan x=z$: $f(t)=\pi^2/\sin t$.
$f(t)=\pi^2/\sin t$. $\int_0^{\pi/2}\sin t\,dt=1$.
Correct Answer: 1