Is this integral using change of variables technique correct?

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Thread starter WMDhamnekar
Start date Jun 4, 2022
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Change Change of variables Integral Multiple integrals Variables

In summary, the change of variables technique is a method used to simplify integrals by substituting a new variable for the original variable. It is used when encountering integrands with complicated expressions or multiple variables. Any variable can be used for the substitution, as long as it is different from the original variable. The steps to solve an integral using this technique are: 1) Identify a suitable substitution, 2) Differentiate the substitution and substitute into the integral, 3) Solve the new integral, and 4) Substitute back in the original variable to get the final answer. To check if the integral is correct, you can differentiate the final answer and compare it to the original integrand.

Jun 4, 2022

WMDhamnekar

MHB

TL;DR Summary: https://youtu.be/TvM_SeMeg9I

Is the answer given above correct?

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Jun 5, 2022

WMDhamnekar

MHB

In my opinion, the calculated answer is correct.

1. What is the change of variables technique in integration?

The change of variables technique in integration is a method used to simplify integrals by substituting a new variable for the original variable in the integrand. This technique is particularly useful for integrals that involve complicated algebraic expressions or trigonometric functions.

2. How do I know when to use the change of variables technique in integration?

You can use the change of variables technique in integration when the integrand contains a function that can be simplified or transformed by substituting a new variable. This can include functions like trigonometric functions, exponential functions, or algebraic expressions.

3. What are the steps for using the change of variables technique in integration?

The steps for using the change of variables technique in integration are as follows:

Choose a new variable to substitute for the original variable in the integrand.
Write the differential of the new variable in terms of the original variable.
Substitute the new variable and its differential into the integrand.
Simplify the integrand and solve the integral using standard integration techniques.
Substitute the original variable back into the final solution.

4. Can the change of variables technique be used for definite integrals?

Yes, the change of variables technique can be used for both indefinite and definite integrals. When using this technique for definite integrals, the limits of integration must also be adjusted to correspond with the new variable.

5. How can I check if my integral using the change of variables technique is correct?

To check if your integral using the change of variables technique is correct, you can differentiate the final solution and see if it matches the original integrand. You can also use a graphing calculator or software to graph the original integrand and the final solution and see if they overlap.