How do you integrate using substitution: the integral from 1 to 3 of: ((3x^2)+(2))/((x^3)+(2x))

Question

How do you integrate using substitution: the integral from 1 to 3 of: ((3x^2)+(2))/((x^3)+(2x))

There is a trick to this one that grealy simplifies the integral.
Let u = x^3 + 2x. Then
du = (3x^2 + 2)dx
The integral then bemoces just the integral of du/u, which is
ln u = ln (x^3 + 2x)
The limits of integration are from 1 to 3, so the answer is
ln (27 + 6) - ln (1+2) = ln 33/3 = ln 11

THANK YOU SOOOOOOOOOOOO MUCH!!! I actually understand that. Thanks!

Answer 1

You're welcome! I'm glad I could help you understand the concept of integrating using substitution. It's a powerful technique that can greatly simplify integrals. If you have any more questions or need further clarification, feel free to ask.