For any integer \(n\),
\[ \int_{0}^{\pi} e^{\cos^2 x} \cdot \cos^3(2n + 1)x \, dx \text{ has the value.} \]

Question

If \( f(x) = \int_{0}^{\sin^2 x} \sin^{-1} \sqrt{t} \, dt \) and \( g(x) = \int_{0}^{\cos^2 x} \sin^{-1} \sqrt{t} \, dt \), then the value of \( f(x) + g(x) \) is:

Answer 1

Step 1: Identify the integrand's odd oscillatory term: \( \cos^3((2n + 1)x) \).
Step 2: Recall that integrating an odd function over a symmetric interval yields zero.
Step 3: Consequently, the integral's value is \( 0 \).

For any integer \(n\),
\[ \int_{0}^{\pi} e^{\cos^2 x} \cdot \cos^3(2n + 1)x \, dx \text{ has the value.} \]

The Correct Option is C

Solution and Explanation

Top Questions on Integration

Questions Asked in WBJEE exam

For any integer \(n\),\[ \int_{0}^{\pi} e^{\cos^2 x} \cdot \cos^3(2n + 1)x \, dx \text{ has the value.} \]

The Correct Option is C

Solution and Explanation

Top Questions on Integration

Questions Asked in WBJEE exam

For any integer \(n\),
\[ \int_{0}^{\pi} e^{\cos^2 x} \cdot \cos^3(2n + 1)x \, dx \text{ has the value.} \]