Definite Integrals by Substitution – Changing the Limits of Integration
Krystian Karczyński
Founder and General Manager of eTrapez.
Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.
He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.
To tackle the above problem on definite integration, we need to use substitution 
, and we’re all on the same page here. But what about the limits of integration?
Integration Limits in Substitution Tasks for Definite Integrals
Can we write:
?
Absolutely not. The issue with integration limits. In the definite integral on the left, they relate to variable
, and on the right to variable 
, so they must also change with the variable.
One approach (which I recommend in my course) is to completely sidestep the problem by solving the indefinite integral (without integration limits) on the side and then inserting the integration limits of 5 and 2 into the result (with variable x).
The other approach is to confront the problem head-on and change the integration limits according to the substitution. Since the limits in variable x are: 2 and 5, after substitution: 
they will be in variable t respectively: 1 and 22, and I obtained these results by substituting 2 and 5 for x in the substitution . So the correct transition would be:
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