Question

If \[ \int_0^2 2e^{2x} \, dx = \int_0^a e^x \, dx, \text{ the value of 'a' is:} \]

• A. 1
• B. 2
• C. 4
• D. \( \frac{1}{2} \)

Hint:

Always simplify integrals step by step, and use substitution effectively for exponential functions. Ensure the limits of integration are updated correctly when changing variables.

Answer 1

The Correct Option is C

Step 1: Solve the left-hand side integral. Evaluate: \[ \int_0^2 2e^{2x} \, dx. \] Using the substitution \( u = 2x \), \( du = 2dx \), and adjusting the limits: \[ \int_0^2 2e^{2x} \, dx = \int_0^4 e^u \, du = e^u \Big|_0^4 = e^4 - 1. \] Step 2: Solve the right-hand side integral. Evaluate: \[ \int_0^a e^x \, dx = e^x \Big|_0^a = e^a - 1. \] Step 3: Equate the two integrals. Equating the left-hand side and right-hand side: \[ e^4 - 1 = e^a - 1. \] Cancel \( -1 \) from both sides: \[ e^4 = e^a. \] Taking the natural logarithm: \[ a = 4. \] Final Answer: \[ \boxed{a = 4.} \]