Integrate ?sec?^3 xdx

Answer : Integral (sec³(x) dx)

Integral (sec(x) sec²(x) dx)

Now, use integration by parts.

Let u = sec(x). dv = sec²(x) dx
du = sec(x)tan(x). v = tan(x) 

Integral (sec³(x) dx) = sec(x)tan(x) - Integral (sec(x)tan²(x) dx)

Using tan²(x) = sec²(x) - 1.

Integral (sec³(x) dx) = sec(x)tan(x) - Integral (sec(x)[sec²(x)
- 1] dx)


Integral (sec³(x) dx) = sec(x)tan(x) - Integral( (sec³(x) - sec(x)) dx)

Now, separate into two integrals.

Integral (sec³(x) dx) = sec(x)tan(x) - [Integral (sec³(x) dx) -
Integral (sec(x) dx)] 


Integral (sec³(x) dx) = sec(x)tan(x) - Integral (sec³(x) dx) + 
Integral (sec(x) dx)
 

 

 

Now move - Integral (sec³(x) dx) to the LHS of equation

2 Integral (sec³(x) dx) = sec(x)tan(x) + Integral (sec(x) dx)

{integeral sec x = ln|sec(x) + tan(x)| }
2 Integral (sec³(x) dx) = sec(x)tan(x) + ln|sec(x) + tan(x)| 

now divide the complete equation by 2

Integral (sec³(x) dx) = (1/2)sec(x)tan(x) + (1/2)ln|sec(x) + tan(x)|


= (1/2)sec(x)tan(x) + (1/2)ln|sec(x) + tan(x)| + C Answer  { C is a constant }