Question

The integral \[ \int_0^{\frac{\pi}{2}} \cos x \cdot e^{\sin x} \, dx \] is equal to:

• A. 0
• B. \( 1 - e \)
• C. \( e - 1 \)
• D. 1

Hint:

When dealing with integrals involving trigonometric functions and exponentials, consider using substitution to simplify the integrand.

Answer 1

The Correct Option is B

The integral is transformed using the substitution \( u = \sin x \), yielding \( du = \cos x \, dx \). The limits of integration change from \( x = 0 \) to \( u = 0 \), and from \( x = \frac{\pi}{2} \) to \( u = 1 \). Consequently, the integral simplifies to: \[ \int_0^1 e^u \, du = e^u \Big|_0^1 = e - 1. \]