Question:
\( \int \frac{x e^x}{(1+x)^2} \, dx = \)
\( \int \frac{x e^x}{(1+x)^2} \, dx = \)
Show Hint
Check reverse differentiation for tricky integrals.
Updated On: May 1, 2026
- \( \frac{e^x}{1+x} + C \)
- \( \frac{e^x}{1+e^x} + C \)
- \( \frac{e^{2x}}{1+x} + C \)
- \( \frac{e^{-x}}{1+x} + C \)
- \( \frac{e^{2x}}{1+x} + C \)
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The Correct Option is A
Solution and Explanation
Step 1: Try derivative of \( \frac{e^x}{1+x} \).
Step 2: Use quotient rule.
\[ \frac{(1+x)e^x - e^x}{(1+x)^2} \]
Step 3: Simplify numerator.
\[ = \frac{x e^x}{(1+x)^2} \]
Step 4: Matches integrand.
Step 5: Hence result.
\[ \frac{e^x}{1+x} + C \]
Step 2: Use quotient rule.
\[ \frac{(1+x)e^x - e^x}{(1+x)^2} \]
Step 3: Simplify numerator.
\[ = \frac{x e^x}{(1+x)^2} \]
Step 4: Matches integrand.
Step 5: Hence result.
\[ \frac{e^x}{1+x} + C \]
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