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