Integration of Exponential Times Sine: Method 2
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We wish to find
Equation 1
Can we do this directly?
No.
Why?
Because we don't know of a function for which the derivative is
How can we solve this problem?
We can try Integration by Parts.
Set that up in general.
Equation 2
How do we choose the parts?
Usually we want any power of x to be part of
so that
will be a smaller power. Since there are no powers here,
we can choose either the exponential or the sine as
.
The rest of the integrand will be
.
First use
Express
.
Express
.
Express
Substitute these results in Equations 1 and 2.
Simplify.
Equation 3
Is this simpler than Equation 1?
No, but it is different. As we'll see, we can again use Integration by Parts.
Let's use a subscript "4" for this, and only work on the integral.
We'll take care of the 3/2 factor later.
Set up Integration by Parts for this remaining integral.
What shall we use for
?
Again, let's use
.
Express the corresponding
Substitute in
Simplify.
Note that the second term is a multiple of the integral we are trying to find. You might think that we are going in circles and that Integration by Parts is not working. It is working, as we'll see.
Substitute this result into Equation 3 to find a new expression for
.
Let’s look closely at this result. The last term is a multiple of
.
Rewrite to indicate this.
Collect the
terms on the left.
Solve for
We get
How can we check this result?
We can check by finding
and comparing it with the integrand in Equation 1.
First, set up the application of the Product Rule. We'll get 4 terms out of the two products.
Carry out the differentiation.
Distribute the multiplication.
Note that the first and fourth terms eliminate. Combine the remaining terms.
Does it check?
Yes, differentiation produces the integrand of Equation 1.
Let's review how Integration by Parts worked in this problem. By applying the method twice we got a result that
was equal to a sum of two terms plus a multiple of itself. We then collected the two terms involving
on one side and solved.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor