Using u substitution, which of the following is equivalent to this integral?

[lude1]

Homework Statement

Using the u substutituion u = 2x + 1, ∫(2x + 1)^1/2dx (when x goes from 0 to 2) is equivalent to?

Answer: (1/2)*∫(u)^1/2du (when x goes from 1 to 5)

Homework Equations

The Attempt at a Solution

If u is 2x + 1, then du = 2dx. Thus, I get

(1/2)*∫(u)^1/2du (when x goes from 0 to 2)​

The only problem is where did the x goes from 1 to 5 come from? Did I do something wrong, or is the AP packet wrong?

 

[Refraction]

For a question like this, when you use substitution to solve a definite integral and have it in a form with u in it, you need to have the two limits in terms of u as well, not x.

So in this example, the limits are 0 to 2 and you know that u = 2x + 1. Just substitute in x = 0 and x = 2, and it gives you the new limits of 1 and 5.

 

[lude1]

Oh, yes, you are right. I forgot that in order for my answer to be right, it had to be (1/2)*∫(2x + 1)^1/2du (when x goes from 0 to 2).

Thanks!

 

[Mark44]

Oh, yes, you are right. I forgot that in order for my answer to be right, it had to be (1/2)*∫(2x + 1)^1/2du (when x goes from 0 to 2).

Thanks!

That should be (1/2)*∫(2x + 1)^1/2dx.

It's not necessary to change the limits of integration when you do a substitution. You can work with the indefinite integral, make the substitution, get your antiderivative, undo your substitution, and then evaluate your antiderivative at the original limits.