Definite Integration
Integral Calculus-2
star_batch_jee_advanced_2025
Grade 12
Question:
If $\int_0^1 \frac{c\sin^3 x}{4-\cos^2 x} dx = \pi\left[1-\frac{a\ln b}{c}\right]$ where $a$ and $b$ are prime and $c \in \mathbb{N}$, then $a+b+c = $ ____.
Step-by-Step Solution
Key Concept: Integration by parts combined with trigonometric substitution reduces a complex rational trigonometric integral to a polynomial form.
The integral $I = \int_{\pi/4}^{\pi/2} x \frac{\cos 2x \cos x}{\sin^3 x} dx$ is evaluated using integration by parts with $u = x$ and $dv = \frac{\cos 2x \cos x}{\sin^3 x} dx$. After computing the antiderivative as $\frac{\cosec^2 x}{2} - \frac{\cosec^6 x}{6}$, the boundary evaluation yields $I = -J$. The remaining integral $-J$ is solved by substitution $\cot x = y$, leading to $J = \frac{16}{45}$. Thus $a + b = 16 + 45 = 61$.
Correct Answer: 10