Integral Calculus-2
Integral Calculus-2
Allen Star Batch
Grade 12
Question:
The value of $\int_0^{\pi/2} \log(\sin^2 \theta + k^2 \cos^2 \theta) d\theta$, where $k \geq 0$, is:
$\pi \log(1 + k) + \pi \log 2$
$\pi \log(1 + k)$
$\pi \log(1 + k) - \pi \log 2$
$\log(1 + k) - \log 2$
Step-by-Step Solution
Key Concept: Differentiate the parameter $k$ inside the logarithm to obtain a rational integrand that may be more tractable than the original logarithmic form.
Given $F(k) = \int_0^{\pi/2} \ln(\sin^2\theta + k^2\cos^2\theta)d\theta$, differentiate with respect to $k$: $F'(k) = \int_0^{\pi/2} \frac{2k\cos^2\theta}{\sin^2\theta + k^2\cos^2\theta}d\theta$. This integral can be evaluated using standard techniques or further substitution to find the derivative, which leads to finding $F(k)$ by integration.
Correct Answer: 3