when figuring intergrals, how do you apply change of variables?

IT depends on what you mean by "figuring".

When working with integrals, sometimes it is helpful to change the variables of integration. This technique is known as a change of variables or substitution. It allows us to transform the integrand into a simpler or more convenient form, making it easier to evaluate the integral.

To apply the change of variables technique, you can follow these steps:

1. Choose a suitable substitution: Look for a substitution that simplifies the integrand. This might involve substituting a single variable with another variable or introducing a new function when dealing with multiple variables.

2. Define the new variable: Once you have chosen a substitution, let the new variable (let's say it is u) be a function of the original variable (let's say x). Determine the relationship between the original and new variables.

3. Calculate the differential: Differentiate the relationship you found in step 2 to find the differential of the new variable, du, in terms of dx.

4. Express the original integral: Rewrite the integral using the new variable and its differential. This step usually involves substituting the original variable with the new variable and multiplying by the appropriate terms.

5. Evaluate the limits of integration: If the integral has definite limits, make sure to express them in terms of the new variable.

6. Evaluate the new integral: After transforming the integral using the change of variables, simplify and evaluate the new integral. Be sure to substitute back the original variable if required.

It is important to choose an appropriate substitution that simplifies the integrand because it can significantly affect the complexity of the resulting integral. Common substitution techniques include u-substitution, trigonometric substitution, and exponential substitution.

Remember, practice and familiarity with different techniques will help you to identify suitable substitutions and apply the change of variables effectively.