For an integral with two functions that don't go away (exponential and trigonometric functions whose derivatives and antiderivatives are exponentials and trigonometric functions respectively), the goal is to get some proportion of the original integral (plus at least two additional functions) to equal the original integral. We can then subtract the former from the latter and get a nice answer.
In this example, with an area of an exponential function and a trigonometric function of the form $f(x)*g(x)dx, we can use integration by parts twice, setting the trigonometric function as dv = trig dx because it will return to some degree of its original form after two rounds of integration by parts.