Question

The integral \(\int e^x \sqrt{e^x} \, dx\) equals:

• A. \(\frac{3}{2} e^x \sqrt{e^x} + C\)
• B. \(\frac{2}{3} e^x \sqrt{e^x} + C\)
• C. \(\frac{5}{2} e^{2x} \sqrt{e^x} + C\)
• D. \(\frac{2}{5} e^{2x} \sqrt{e^x} + C\)
• E. \(\frac{2}{3} e^{2x/3} + C\)

Hint:

Always simplify the expression before integrating, especially with exponents. It often reduces the integral to a basic form that is straightforward to solve.

Answer 1

The Correct Option is B

First, simplify the integrand: \[ e^x \sqrt{e^x} = e^x \cdot e^{x/2} = e^{3x/2} \] Now, integrate the simplified expression: \[ \int e^{3x/2} \, dx \] Let \(u = \frac{3x}{2}\), then \(dx = \frac{2}{3} du\). Substitute and integrate: \[ \int e^u \cdot \frac{2}{3} \, du = \frac{2}{3} \int e^u \, du = \frac{2}{3} e^u + C \] Substitute back for \(x\): \[ = \frac{2}{3} e^{3x/2} + C \] Since \(e^{3x/2} = e^x \sqrt{e^x}\), we can rewrite the integral as: \[ = \frac{2}{3} e^x \sqrt{e^x} + C \]