# Solving Improper Integral: 2-x^4 |U Substitution

• Mathematics07
In summary, the conversation discusses the process of evaluating the integral from -infinity to infinity of (2-x^4)dx using u-substitution. However, it is determined that u-substitution is not necessary in this case and the improper integral can be evaluated by taking the limit as s approaches infinity of the integral from -s to s. This involves finding the anti-derivative and substituting in the infinities before evaluating.

## Homework Statement

The integral from -infinity to infinity of (2-x^4)dv

U substitution

## The Attempt at a Solution

Dont know what to use as my "u" ?

You have a dv in your given integral. Is this the integral you mean?
$$\int_{-\infty}^\infty (2-x^4) \,dx$$
If so, you do not need u-substitution here. Also, to compute an improper integral of this form you need to evaluate:
$$\lim_{s\to\infty} \int_{-s}^s (2-x^4) \,dx$$

yes that the write integral . . so basically i just need to do regular anti derivate and then evaluate it ?

n!kofeyn said:
$$\int_{-\infty}^\infty (2-x^4) \,dx$$
If so, you do not need u-substitution here. Also, to compute an improper integral of this form you need to evaluate:
$$\lim_{s\to\infty} \int_{-s}^s (2-x^4) \,dx$$

so for an example like this you would just do a regular anti derivative and then substitute your "s" in for the infinities and then evaluate >?

-Jay J-

## 1. What is an improper integral?

An improper integral is an integral where either the upper or lower limit of integration is infinite or the integrand has a singularity within the interval of integration. This means that the area under the curve cannot be calculated using the standard methods of integration and requires alternative techniques.

## 2. What is the U-Substitution method?

The U-Substitution method, also known as the "change of variables" method, is a technique used to simplify integrals by substituting a new variable for the existing variable. The new variable, u, is chosen in such a way that it cancels out with the existing variable in the integrand, making the integration process easier.

## 3. How do you solve an improper integral using U-Substitution?

To solve an improper integral using U-Substitution, follow these steps:

1. Identify the part of the integrand that can be substituted with a new variable.

2. Use the substitution u = g(x) to replace the existing variable with the new variable.

3. Rewrite the integral in terms of u.

4. Solve the integral with respect to u.

5. Finally, substitute back for u in the final answer.

## 4. What is the role of the 2-x^4 term in this improper integral?

The 2-x^4 term is the integrand in this improper integral. It represents the function whose area under the curve we are trying to calculate. In this case, it is being raised to the power of -1, indicating an inverse function.

## 5. Are there any special cases to consider when solving improper integrals using U-Substitution?

Yes, there are a few special cases to consider when using U-Substitution:

1. When the limits of integration are infinite, the substitution u = 1/x may be used.

2. When the integrand contains a radical, the substitution u = g(x) = √x may be used.

3. When the integrand contains a trigonometric function, the substitution u = g(x) = tan(x) or u = g(x) = sec(x) may be used.