Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade None
Question:
The value of $a$ for which the equation $\int_0^a \sin^2\left(\frac{t}{2}\right)dt = a^2x^2 - \frac{1}{2}(3x-1) + \frac{1}{a^2}$ possesses a solution are:
$\pm\frac{1}{\sqrt{2n\pi}}, n \in \mathbb{N}$
$\pm\frac{1}{\sqrt{2n\pi - \pi/2}}, n \in \mathbb{N}$
$\pm\frac{1}{\sqrt{n\pi + \pi/2}}, n \in \mathbb{N}$
None of these
Step-by-Step Solution
Key Concept: Repeated integration by parts combined with trigonometric identities creates a solvable system of integrals.
Integration by parts is applied to $I_{4,3} = \int \cos^3 x \sin 3x dx$. Using the identity $\sin x \cos 3x = -\sin 2x + \sin 3x \cos x$, the integral is reduced to a combination of $I_{4,3}$, $I_{3,2}$, and lower-order terms. Solving the resulting relation $\frac{7}{3}I_{4,3} - \frac{4}{3}I_{3,2} = \frac{\cos 3x \cos^3 x}{3} + C$ yields the final answer.
Correct Answer: 3