Question

$\displaystyle \int \left(\frac{1}{(1+x)^2} - \frac{2}{(1+x)^3}\right)e^x \, dx$ is equal to:

• A. $-\frac{2}{(1+x)^2} + C$
• B. $\frac{1}{(1+x)^2} + C$
• C. $\frac{2}{(1+x)^2} + C$
• D. $-\frac{2e^x}{(1+x)^2} + C$
• E. $\frac{e^x}{(1+x)^2} + C$

Hint:

Always check if integrand is derivative of a known expression.

Answer 1

Answer: (E)

Concept:
• Recognize derivative form of $\frac{e^x}{(1+x)^2}$

Step 1: Let function be
\[ f(x) = \frac{e^x}{(1+x)^2} \]

Step 2: Differentiate
\[ f'(x) = \frac{e^x}{(1+x)^2} - \frac{2e^x}{(1+x)^3} \]

Step 3: Match integrand
\[ f'(x) = \left(\frac{1}{(1+x)^2} - \frac{2}{(1+x)^3}\right)e^x \] Final Conclusion:
\[ \int = \frac{e^x}{(1+x)^2} + C \]