Question

Evaluate the integral: $ \int e^{-x} \cdot e^{3x} \, dx $

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

Hint:

When combining exponentials, remember that the exponents add together. Always check the integral of exponential functions using their basic properties.

Answer 1

The Correct Option is B

The given integral is: \[ \int e^{-x} \cdot e^{3x} \, dx \] Using the property of exponents that \( e^a \cdot e^b = e^{a + b} \), we can combine the exponents: \[ \int e^{-x + 3x} \, dx = \int e^{2x} \, dx \] Now, integrate \( e^{2x} \) with respect to \( x \): \[ \int e^{2x} \, dx = \frac{e^{2x}}{2} + C \] 
Thus, the correct answer is \( \frac{e^{2x}}{2} + C \).