If I understand this correctly we have

This can be viewed as a function composition. Let

Then

Now by the chain rule

Remember that the fundemental theorem of calculus says that(Notice the upper limit of integration) check you book for the hypothsis

Combining these two results we get

Notice that if you integrate both sides of the last equaion with the same u substition that we did above you will get the same function g back. i.e

I hope this helps