\(\int \displaystyle\frac{xe^x}{(x+1)^2}\, dx=\)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{e^x}{x+1}+c\)
Writing the numerator x as (x + 1) − 1 splits the fraction into two parts:
\( \displaystyle \frac{x}{(x+1)^2} = \frac{(x+1)-1}{(x+1)^2} = \frac{1}{x+1} - \frac{1}{(x+1)^2} \)
With \(f(x) = \frac{1}{x+1}\), the second term is \(f'(x) = -\frac{1}{(x+1)^2}\). So the integral has the standard form
\( \displaystyle \int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + c \)
Hence
\( \displaystyle \int\frac{xe^x}{(x+1)^2}\,dx = \frac{e^x}{x+1} + c \)
NCERT (2026-27 edition):
Integration by Parts
