Integrate (sec(2x))^3*tan (2x)dx

Question

How do you do this?

Answer 1

First, we expand the integral.  This will allow us to find the relationship between the terms.

 

∫sec²(2x)sec(2x)tan(2x)dx

 

 

Now we can use u-substitution.

 

Let       u = sec(2x)

         du = 2sec(2x)tan(2x)dx

 

        (1/2)du = sec(2x)tan(2x)dx

 

 

Substituting these values of u into the integral, we get

 

∫u²(1/2)du

 

 

Now the integral is easy to evaluate using the power rule.

 

(1/2)∫u² du =

 

(1/6)u³ + C

 

 

Now, we substitute the term represented by the variable in the integral.

 

(1/6)sec³(2x) + C

 

 

To check, if you take the derivative of this solution, you get back the expression we need to integrate.