Definite Integration
King's Property
nta_pyq_2024_apr
Grade 12
Question:
If the value of the integral $\int_{-1}^{1}\dfrac{\cos\alpha x}{1+3^x}\,dx$ is $\dfrac{2}{\pi}$, then a value of $\alpha$ is:
$\dfrac{\pi}{3}$
$\dfrac{\pi}{6}$
$\dfrac{\pi}{4}$
$\dfrac{\pi}{2}$
Step-by-Step Solution
Key Concept: Use King's property: $I+I'=\int_{-1}^1\cos\alpha x\,dx$. Adding gives $2I=\frac{2\sin\alpha}{\alpha}$, so $I=\frac{\sin\alpha}{\alpha}=\frac{2}{\pi}$.
$I=\frac{\sin\alpha}{\alpha}=\frac{2}{\pi}\Rightarrow\alpha=\pi/2$.
Correct Answer: 4