`int cos^2(3x) dx` Find the indefinite integral

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Recall  that indefinite integral follows `int f(x) dx = F(x) +C ` where:

`f(x) ` as the integrand function

`F(x)` as the antiderivative of f(x)

`C` as the constant of integration..

 For the given integral problem: `int cos^2(3x) dx` , we can evaluate this by using a trigonometric identity. Recall that:

`cos^2(theta) = (1 + cos(2theta))/2` .

Applying the trigonometric identity, we get:

`int cos^2(3x) dx = int (1 + cos(2* 3x))/2 dx`

                              `= int ( 1 + cos(6x))/2dx`

                              `=int ( 1/2 + cos(6x)/2)dx`

                        

Apply the basic integration property: : `int (u+v) dx = int (u) dx + int (v) dx` .

`int ( 1/2) + cos(6x)/2)dx =int ( 1/2) dx + int cos(6x)/2dx`

For the first integral: `int (1/2) dx` , we may apply basic integration property: `int c dx = cx` .

`int (1/2) dx = 1/2x or x/2`

For the second integral:  `int cos(6x)/2dx` , we  may apply basic integration property: `int c f(x) dx = c int f(x) dx` .

`1/2 int cos(6x) dx` .

Apply u-substitution by letting `u = 6x` then `du = 6 dx` or `(du)/6 = dx` .

`1/2 int cos(6x) dx = 1/2 int cos(u) * (du)/6`

                               `= 1/2*1/6 int cos(u) du`

                               `= 1/12 sin(u)`  

Plug-in `u = 6x` on `1/12sin(u)` , we get:

`1/2 int cos(6x) dx = 1/12 sin(6x) ` or `sin(6x)/12`

Combining the results, we get the indefinite integral as:

`int cos^2(3x) dx = x/2 + sin(6x)/12+C`

 

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial