Question:
Evaluate $\int \frac{x-1}{(x+1)^3}e^x dx$:
Evaluate $\int \frac{x-1}{(x+1)^3}e^x dx$:
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Match $e^x[f(x)+f'(x)]$ structure quickly.
Updated On: Jun 10, 2026
- $\frac{e^x}{(x+1)^2}+c$
- $\frac{-e^x}{(x+1)^2}+c$
- $\frac{2e^x}{x+1}+c$
- $\frac{-e^x}{(x+1)^4}+c$
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The Correct Option is A
Solution and Explanation
Rewrite:
\[
x-1=(x+1)-2
\]
\[
I=\int e^x\left[\frac{1}{(x+1)^2}-\frac{2}{(x+1)^3}\right]dx
\]
Let:
\[
f(x)=\frac{1}{(x+1)^2}
\]
\[
I=e^x f(x)+c
\]
\[
I=\frac{e^x}{(x+1)^2}+c
\]
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