First make a substitution and then use integration by parts to evaluate the integral

We need to make a substitution then use integration by parts.

Let us make the substitution:

so:

therefore

so our equation can be changed.

Now use integration by parts.

Let

We will call that equation 1.

...

We need to make a substitution then use integration by parts.

Let us make the substitution:

so:

therefore

so our equation can be changed.

Now use integration by parts.

Let

We will call that equation 1.

Now we need to evaluate that second integral with integration by parts again.

Now let us plug this result for int e^t cos(t) dt back into equation 1.

add to both sides:

sub back in our original or

divide both sides by 2 and add the constant of integration. And were done!!!!