Step-by-step Solutions for Definite Integrals in Wolfram|Alpha
One of the most popular queries on Wolfram|Alpha is for definite integrals. So we’re especially excited to announce that Step-by-step solutions for these are now available! The general method used to find the steps for definite integrals is to tap into the already existing “Show steps” functionality for indefinite integration, and then to use the fundamental theorem of calculus.
Now you may think it was trivial to add this functionality given that indefinite integrals already have steps, but there are many tricky cases to consider: before we even begin to integrate, the continuity of the function is examined. If there are discontinuities in the integration domain, the domain is split and the integral is evaluated separately over each domain.
We must determine if the integral is proper or improper.
Absolute values need to be handled carefully.
Symmetries can be exploited.
Simplification of radicals and logarithms must be done very carefully.
Finally, the fundamental theorem of calculus requires that the antiderivative is continuous over the integration domain (see this blog post for more information). Therefore, we need to be careful when finding the indefinite integral, and always ensure the result will be continuous. One way to do this is to detect when we will have a discontinuous antiderivative and split the integration domain up.
Integration is an extremely nontrivial problem, so we hope these Step-by-step solutions will help you learn how they can be done. Be sure to check out Step-by-step solutions for other topics too.