Integrals using u substitution

In summary, u substitution is a method used to simplify integrals by replacing the original variable with a new variable, u. It is typically used when the integral contains a function within a function, and involves several steps such as identifying the function to be replaced and solving the integral using u as the new variable. While u substitution may not always be the best method for solving integrals, it is a useful tool to have in your mathematical toolkit. However, there may be limitations to its use, such as not working for all types of integrals or leading to more complex solutions. It is important to check the solution to ensure it is equivalent to the original integral.
  • #1
howsockgothap
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Homework Statement



Using substitution, find the integral of 32x2/(2x+1)3

Homework Equations





The Attempt at a Solution



I initially tried plugging u in for 32x2 but that wouldn't work because it won't cancel out with the problem below it anyway. I'm pretty sure we are not expected to multiply the bottom function out, so how would I go about doing this?
 
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  • #2
use partial fractions
 
  • #3
Just use u=2x+1. You would have to do it anyway after partial fractions. So x=(u-1)/2. Then just expand the numerator.
 

1. What is u substitution in integrals and why is it used?

U substitution is a method used to simplify the integration process by substituting a new variable, u, for the original variable in an integral. This is done to make the integral easier to solve, as it can often be transformed into a more familiar form.

2. How do I know when to use u substitution in integrals?

U substitution is typically used when the integral contains a function within a function, such as f(g(x)). In these cases, u substitution can be used to simplify the integral by replacing the inner function with u. It may also be used when the integral contains trigonometric functions or exponential functions.

3. What is the process for using u substitution in integrals?

The process for using u substitution in integrals involves the following steps:

  1. Identify the function within the integral that can be replaced with u.
  2. Find the derivative of u with respect to the original variable.
  3. Substitute the function and its derivative with u and du in the integral.
  4. Solve the integral using u as the new variable.
  5. Substitute u back in for the original variable to get the final solution.

4. Can u substitution be used for all types of integrals?

No, u substitution may not always be the best method for solving integrals. It is most commonly used for integrals containing a function within a function, but there may be other methods that are more efficient for other types of integrals.

5. Are there any limitations to using u substitution in integrals?

Yes, there are some limitations to using u substitution in integrals. It may not work for all types of integrals, and there may be cases where the substitution leads to a more complex integral. It is important to check your solution and make sure it is equivalent to the original integral.

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