When we make a substitution in a definite integral in the form

we need to remember that the bounds are from x=a to x=b:

We then have two choices:

1. Plug x=a and x=b into u to find the bounds in terms of u.

2. Plug u(x) back in and use the bounds a→b.

Example

Consider ∫13​x2+12x​dx. We notice 2x is the derivative of x2+1, so we let u=x2+1⇒dxdu​=2x. Thus the integral becomes:

Now we have two choices:

1. Plug x=3 and x=1 into u=x2+1, giving u=10 and u=2. Then

   [ln∣u∣]x=1x=3​=[ln∣u∣]210​=ln10−ln2=ln5

2. Replace u with x2+1:

   [ln∣u∣]x=1x=3​=[ln∣∣​x2+1∣∣​]13​=ln10−ln2=ln5