Integration of xSinCos
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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 this integrand is the derivative.
Isn't the integral equal to
?
No, because of the factor x.
How can we solve this problem?
We can try to find a trig identity which will simplify the integrand.
There is one available on this page: Trig Cheat Sheet.
See if you can find it.
Look in the Double Angle section.
We can use
.We simply have to move the factor of 2 to the other side.
How do we define
here?
We take
Use this information to rewrite Equation 1 in terms of just the sine function.
Equation 1 becomes
Equation 2
How does this help?
Now we can try Integration by Parts.
Set that up in general.
Equation 3
How do we choose the parts?
Usually we want the most complicated part of the integrand to be
so that
will be simpler. The rest of the integrand will be
.
Which factor is the most complicated here?
Powers of x are usually the most complicated and get simpler in the transition from
Express
.
Express
.
Express
.
Express
Substitute these results in Equations 2 and 3.
Is this simpler than Equation 2?
Yes, it contains only a trig function in the integral.
Simplify the integral.
We get
Complete the integration.
How can we check this result?
We can check by finding
and comparing it with the integrand in Equation 2.
First, set up the application of the Product Rule.
Carry out the differentiation.
Simplify.
Does it check?
Yes, differentiation produces the integrand of Equation 2.
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