Definite Integration
Integral Equation with Trigonometric Substitution — Finding α
nta_pyq_2023_apr
Grade 12
Question:
If $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $\displaystyle\int_0^{\pi/2}f(\sin 2x)\sin x\,dx+\alpha\int_0^{\pi/4}f(\cos 2x)\cos x\,dx=0$, then the value of $\alpha$ is
Step-by-Step Solution
Key Concept: Split $\int_0^{\pi/2}f(\sin2x)\sin x\,dx$ into two halves and apply substitution on the second half. Show it equals $\sqrt{2}\int_0^{\pi/4}f(\cos2x)\cos x\,dx$.
$\int_0^{\pi/2}=\sqrt{2}\int_0^{\pi/4}f(\cos2x)\cos x\,dx$. Sum$=0\Rightarrow\alpha=-\sqrt{2}$.
Correct Answer: 4