\(\displaystyle\int \frac{(x - 3)e^x}{(x - 1)^3}\, dx\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{e^x}{(x - 1)^2} + C\), C is an arbitrary constant
Writing x − 3 as (x − 1) − 2 splits the fraction:
\( \displaystyle \frac{x - 3}{(x - 1)^3} = \frac{(x - 1) - 2}{(x - 1)^3} \)
\( \displaystyle \frac{x - 3}{(x - 1)^3} = \frac{1}{(x - 1)^2} - \frac{2}{(x - 1)^3} \)
With \(f(x) = \frac{1}{(x - 1)^2}\), \(f'(x) = -\frac{2}{(x - 1)^3}\), so the integrand has the form \(e^x[f(x) + f'(x)]\). By the standard result \(\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C\):
\( \displaystyle \int \frac{(x - 3)e^x}{(x - 1)^3}\,dx = \int e^x\left[\frac{1}{(x - 1)^2} - \frac{2}{(x - 1)^3}\right] dx \)
\( \displaystyle \int \frac{(x - 3)e^x}{(x - 1)^3}\,dx = \frac{e^x}{(x - 1)^2} + C \)
NCERT (2026-27 edition):
Integration by Parts
