Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
If $a$ is a positive integer, then the number of values of $a$ satisfying $$\int_0^{\pi/2} \left[a^2\left(\frac{\cos 3x}{3} + \cos x\right) + a\sin x - 20\cos x\right] dx \leq -\frac{a^2}{3}$$ is:
Step-by-Step Solution
Key Concept: Recognizing that the expression under the square root is a perfect square $(x - \frac{1}{x})^2$ allows direct simplification.
Substitute $x^2 + \frac{2}{x^2} + 1 = t^2$ where $t = x - \frac{1}{x}$, giving $\frac{dt}{2} = (x + \frac{1}{x})dx$. This transforms the integral to $\int \frac{dt}{2\sqrt{t}} = \sqrt{t} + C = \sqrt{x^2 + \frac{2}{x^2} + 1} + C$.
Correct Answer: 2