Question

Find the integral $\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx$.

• A. $0$
• B. $1$
• C. $2$
• D. $\infty$

Answer 1

The correct option is B $1$

We can see that the given integral is an improper integral as one of its limits is not finite.
In such cases where we have to deal with infinity as the limits of definite integral, we’ll change the limit which is not finite to a variable and then put the limits.
$\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx=\underset{a\to \infty }{lim}\underset{0}{\overset{a}{\int }}{e}^{-x}dx$
$=\underset{a\to \infty }{lim}{(-e^{-x})}_0^a$
$=\underset{a\to \infty }{lim}(-e^{-a}-(-e^0\left)\right)$
$=0-(-1)$
$=1$