Chapter 5: Q. 19 (page 477)

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{- \infty}^{\infty} \frac{e^{- x^{2}}}{x^{2}} d x$

Short Answer

Expert verified

The integral is,

$\lim_{x \to -} \int_{1}^{- 1} \frac{e^{- x^{2}}}{x^{2}} d x + \int_{- 1}^{0} \frac{e^{- x^{2}}}{x^{2}} d x + \int_{0}^{1} \frac{e^{- x^{2}}}{x^{2}} d x + \lim_{θ \to \infty} \int_{1}^{8} \frac{e^{- x^{2}}}{x^{2}} d x .$

Step by step solution

Step 1. Given information.

We are given an integrals,

$\int_{- \infty}^{\infty} \frac{e^{- x^{2}}}{x^{2}} d x$

Step 2. Graph of function

Let $f (x) = \frac{e^{- x^{2}}}{x^{2}}$ ,

The graph is as follows,

Step 3. Expressing the Integral.

The function $f (x) = \frac{e^{- x^{2}}}{x^{2}}$ is continuous everywhere except at $x = 0$ .

Thus, split the interval $(- \infty, \infty)$ at $x = 0$ .

Rewriting the improper integral by splitting the interval $(- \infty, \infty)$ at $x = 0$ .

$\int_{- \infty}^{\infty} \frac{e^{- x^{2}}}{x^{2}} d x = \int_{- \infty}^{0} \frac{e^{- x^{2}}}{x^{2}} d x + \int_{0}^{\infty} \frac{e^{- x^{2}}}{x^{2}} d x$

Further, split the intervals at the points $x = \pm 1$ to make the calculations of the integrals more simple way.

Hence, split the interval $(- \infty, \infty)$ as the following subintervals:

$(- \infty, - 1), (- 1, 0), (0, 1) and (1, \infty) .$

Therefore, the improper integral becomes,

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Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{- \infty}^{\infty} \frac{e^{- x^{2}}}{x^{2}} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

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Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.∫-∞∞e-x2x2dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Pure Maths

Probability and Statistics

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{- \infty}^{\infty} \frac{e^{- x^{2}}}{x^{2}} d x$