Question

The value of $ \int {(x - 1){e^{ - x}}dx} $ is equal to
A. $ - x{e^x} + C $
B. $ x{e^x} + C $
C. $ - x{e^{ - x}} + C $
B. $ x{e^{ - x}} + C $

Answer 1

Hint:Here, separate the given integral in two terms. Integrate the first term of a given function using product rule. We will see that the two integrals will be cancelled out, and we are left with algebraic terms only. There is no need to solve the two integrals as both are equal and opposite in signs, so whatever be their values it will be cancelled out.

Complete step by step explanation:
Let $ I = \int {(x - 1){e^{ - x}}dx} $
On separating the terms, we have
 $ I = \int {x{e^{ - x}}dx} - \int {{e^{ - x}}dx} $
For first term, using product rule
 $ I = - x{e^{ - x}} - \int {1 \cdot \left( - \right)} {e^{ - x}}dx - \int {{e^{ - x}}dx + C} $
On rewriting the terms, we have
 $ I = - x{e^{ - x}} + \int {{e^{ - x}}} - \int {{e^{ - x}}dx + C} $
Here, two integrals get cancelled out and we are only left with algebraic terms which do not requires any further integration.
 $ \Rightarrow I = - x{e^{ - x}} + C $ ,where C is the constant of integration.

Hence, option (C) is correct.

Note: In these types of questions, be careful as one given term is exponential. We know that integration of simple exponential functions remains unchanged. Mostly the questions of these types do not require any integration. By simplifying the terms and using product rule of integration or quotient rule of integration, most of the terms get cancelled out without even solving their integrals and we are left with the algebraic terms only. In integration questions, first separate the terms given in function and then check whether integration should be done first to minimize the terms. In many questions where
algebraic term and exponential function is in product form, they get cancelled out, so carefully do the step and mostly focus on terms which get cancelled with each other.