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 $\left(-\infty ,\infty \right)$, then for any real number c,

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

Short answer

$\begin{array}{rcl}& & \underset{d\to -\infty }{\lim }{\int }_d^{c}f\left(x\right)dx+\underset{e\to \infty }{\lim }{\int }_c^{e}f\left(x\right)dx\end{array}$

Step-by-step solution

Step 1. Given information.

Consider the given integral,

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

Step 2. Defining improper integrals.

Since the given function is continuous on the given interval $\left(-\infty ,\infty \right)$. As c is any real number then break the integral interval to rewrite it.

$\begin{array}{rcl}{\int }_{-\infty }^{\infty }f\left(x\right)dx& =& {\int }_{-\infty }^{c}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx\\ & =& \underset{d\to -\infty }{\lim }{\int }_d^{c}f\left(x\right)dx+\underset{e\to \infty }{\lim }{\int }_c^{e}f\left(x\right)dx\end{array}$