Question

$\int e^{x}[\frac{1}{1+x}-\frac{1}{(1+x)^{2}}]dx=$ ________.

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

Hint:

Always look for $f(x) + f'(x)$ when you see $e^x$ in an integral.

Answer 1

The Correct Option is A

Step 1: Concept
Use the property $\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$.

Step 2: Meaning
Let $f(x) = \frac{1}{1+x} = (1+x)^{-1}$.

Step 3: Analysis
Then $f'(x) = -1(1+x)^{-2} = -\frac{1}{(1+x)^2}$. The integral matches the form exactly.

Step 4: Conclusion
The result is $e^x f(x) + C = \frac{e^{x}}{1+x} + C$. Final Answer: (A)