Answer:

Step-by-step explanation:
Given

Required
Evaluate

Rewrite as:

In trigonometry:

Divide both sides by 2



Substitute
for 



Let 
Differentiate:

Make
the subject

Substitute
for

Substitute 2x for u



At this point, we apply the reduction formula:
Which is:

Let n = 2; So, we have:




So, we have:

Integrate 1 with respect to u


Recall that:

So, we have:
![\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}[ \frac{1}{2}u - \frac{cos(u)sin(u)}{2}\du]](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7Bcos%28x%29%5C%20sin%282x%29%5C%20sin%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B4%7D%5B%20%5Cfrac%7B1%7D%7B2%7Du%20-%20%5Cfrac%7Bcos%28u%29sin%28u%29%7D%7B2%7D%5Cdu%5D)
Open bracket

Recall that:
and

So, the expression becomes:


Add constant c

----------------------------------------------------------------------------------------
In trigonometry:

Divide both sides by 2


Replace 2x with 


----------------------------------------------------------------------------------------
becomes



The solution can be further simplified as:
