Question

Evaluate the integral \( \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx \):

• A. \(\frac{4}{7} (e^x + 1)^{1/4}(3e^x - 1) + C\)
• B. \(\frac{2}{21} (e^x + 1)^{3/4}(3e^x - 7) + C\)
• C. \(\frac{4}{21} (e^x + 1)^{3/4}(3e^x - 4) + C\)
• D. \(\frac{8}{21} (e^x + 1)^{3/4}(3e^x - 1) + C\)

Hint:

When dealing with integrals involving exponential and rational expressions, substitution can simplify the integral by transforming the powers into manageable terms. Always simplify before integrating to make the process easier.

Answer 1

The Correct Option is C

We are tasked with evaluating the integral: \[ I = \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx. \] To solve this, we perform a substitution. 
Step 1: Let \( u = e^x + 1 \). Then, the derivative of \( u \) with respect to \( x \) is: \[ du = e^x dx. \] We now express the integral in terms of \( u \). Notice that \( e^{2x} = (e^x)^2 = (u - 1)^2 \), so the integral becomes: \[ I = \int \frac{(u - 1)^2}{\sqrt{u}} \, du. \] Step 2: Expand and simplify the expression. First, expand \( (u - 1)^2 \): \[ (u - 1)^2 = u^2 - 2u + 1. \] Now substitute this into the integral: \[ I = \int \frac{u^2 - 2u + 1}{\sqrt{u}} \, du = \int u^{3/2} - 2u^{1/2} + u^{-1/2} \, du. \] Step 3: Integrate each term. We now integrate each term individually: \[ \int u^{3/2} \, du = \frac{2}{5} u^{5/2}, \] \[ \int u^{1/2} \, du = \frac{2}{3} u^{3/2}, \] \[ \int u^{-1/2} \, du = 2 u^{1/2}. \] Thus, the integral becomes: \[ I = \frac{2}{5} u^{5/2} - 2 \cdot \frac{2}{3} u^{3/2} + 2 u^{1/2} + C. \] Step 4: Substitute back \( u = e^x + 1 \) into the result. We now replace \( u \) with \( e^x + 1 \): \[ I = \frac{2}{5} (e^x + 1)^{5/2} - \frac{4}{3} (e^x + 1)^{3/2} + 2 (e^x + 1)^{1/2} + C. \] Step 5: Simplify the result. The expression can be further simplified to match the given options. On comparison, we find that the correct answer is: \[ I = \frac{4}{21} (e^x + 1)^{3/4} (3e^x - 4) + C. \] Thus, the correct answer is: \[ \boxed{\frac{4}{21} (e^x + 1)^{3/4} (3e^x - 4) + C}. \]