Integrating a Product of Exponential and Trigonometric Functions

Products of functions can often be integrated by parts or using a suitable substitution to obtain a sipler function which can be integrated. Sometimes it happens that the integration by parts method does not result in a simpler integral, but the integral can still be evaluated. A very good example of this is the integral

This function can be integrated by parts:

TakeandthenandSubstituting into (1) gives


is not any easier to integrate thanbut look what happens when we integrate again, by parts.

To integratetakeandthenand

Subsituting this into (2) gives

Addingto both sides gives

so that