Question

The value of \( \frac{d}{dx} \left( \int 2 \sin x \sin x^2 e^{t^2} \right) \) at \( t = \pi \) is:

• A. -1
• B. 1
• C. -2
• D. 2

Hint:

When differentiating integrals, apply Leibniz's rule to account for the derivative of the integrand with respect to the variable and limits of integration.

Answer 1

The Correct Option is C

We are given the integral \( \int 2 \sin x \sin x^2 e^{t^2} \) and asked to differentiate it with respect to \( x \). We need to evaluate the derivative at \( t = \pi \).

Step 1: Use Leibniz's rule for differentiation of an integral.
To differentiate the given integral with respect to \( x \), we apply Leibniz's rule: \[ \frac{d}{dx} \left( \int 2 \sin x \sin x^2 e^{t^2} \right) \] Since the integrand includes terms dependent on \( x \) and \( t \), we must differentiate the expression accordingly. The calculation involves applying the chain rule and evaluating at \( t = \pi \).

Step 2: Evaluate at \( t = \pi \).
After differentiating and substituting \( t = \pi \), the value of the expression is \( -2 \), hence the correct answer is \( -2 \).