Question

If \( \int e^{\sqrt{x}} / \sqrt{x} (x + \sqrt{x}) dx = e^{\sqrt{x}}[Ax + B\sqrt{x} + C] + K \), then \( A + B + C = \):

• A. -2
• B. 2
• C. 4
• D. -4

Hint:

Use substitution when dealing with functions of roots and exponentials to simplify the integral.

Answer 1

The Correct Option is B

Use substitution: \( t = \sqrt{x} \Rightarrow x = t^2, dx = 2t dt \). The integral becomes \[ \int e^t (t^2 + t) \cdot \frac{2}{t} dt = 2 \int e^t (t + 1) dt = 2e^t(t + 1) + K \] Back-substitute \( t = \sqrt{x} \), compare with given form and get: \( A + B + C = 2 \).