Question

Defining improper integrals: Fill in the blanks, using limits and proper definite integrals to express each of the following types of improper integrals.

If f is a continuous on $\left(-\infty ,b\right]$, then

${\int }_{-\infty }^{b}f\left(x\right)dx=\_\_\_\_\_\_$

Short answer

${\int }_{-\infty }^{b}f\left(x\right)dx=\underset{k\to -\infty }{\lim }{\int }_k^{b}f\left(x\right)dx$

Step-by-step solution

Step 1. Given information.

Consider the given integral,

${\int }_{-\infty }^{b}f\left(x\right)dx$

Step 2. Defining improper integrals.

Since the given function is continuous on the given interval $\left(-\infty ,b\right]$, then,

localid="1652093197157" ${\int }_{-\infty }^{b}f\left(x\right)dx=\underset{k\to -\infty }{\lim }{\int }_k^{b}f\left(x\right)dx$

Where kis any real number.