Find the indefinite integral

Indefinite integrals are written in the form of

 where: as the integrand

          as the anti-derivative function 

             as the arbitrary constant known as constant of integration

For the given problem:  has a integrand in a form of trigonometric function. To evaluate this, we apply the identity:

The integral becomes:

Indefinite integrals are written in the form of

 where: as the integrand

          as the anti-derivative function 

             as the arbitrary constant known as constant of integration

For the given problem:  has a integrand in a form of trigonometric function. To evaluate this, we apply the identity:

The integral becomes:

 Apply the basic properties of integration: .

 Apply the basic integration property: .

Then apply u-substitution to be able to apply integration formula for cosine function: .

For the integral: int cos(6x+2x) dx, we let then or .

                                 

                                

                                 

Plug-in on  , we get:

 For the integral:  , we let then or .

                                

                               

                               

Plug-in on  , we get:

Combing the results , we get the indefinite integral as:

or