Question

Let \[ f(t)=\int_{0}^{t} e^{x^2}\Big((1+2x^2)\sin x+x\cos x\Big)\,dx. \] Then the value of \(f(\pi)-f\!\left(\frac{\pi}{2}\right)\) is equal to:

• A. \(-\pi e^{\pi^2/4}\)
• B. \(-\dfrac{\pi}{2}e^{\pi^2/4}\)
• C. \(\dfrac{\pi}{2}e^{\pi^2/4}\)
• D. \(\pi e^{\pi^2/4}\)

Hint:

Whenever you see \(e^{x^2}\) multiplied by algebraic–trigonometric terms, try expressing the integrand as the derivative of \(e^{x^2}\) times a simple function.

Answer 1

The Correct Option is C