let u = sec x dv = sec^2 x
then du = sec x tan x v = tan x
so int(sec^3 x dx) = sec x tan x - int(sec x tan^2 x dx)
= sec x tan x - int(sec x (sec^2 x - 1) dx)
= sec x tan x - int(sec^3 x dx) + int(sec x dx)
= sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C
so now we have
int(sec^3 x dx) = sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C
solve for int(sec^3 x dx):
2 int(sec^3 x dx) = sec x tan x + ln (sec x + tan x) + C
int(sec^3 x dx) = 1/2 (sec x tan x + ln (sec x + tan x)) + C
note that the end result C is different from the above C, but since it's an arbitrary constant it doesn't matter
or
This is one of those integration by parts questions which loop around and you end up getting your answer. Before we start, it's worth remembering that Integral (sec(x) dx) = ln|sec(x) + tan(x)|.
Integral (sec^3(x) dx)
First, split sec^3(x) into sec(x) and sec^2(x).
Integral (sec(x) sec^2(x) dx)
Now, use integration by parts.
Let u = sec(x). dv = sec^2(x) dx
du = sec(x)tan(x). v = tan(x)
Integral (sec^3(x) dx) = sec(x)tan(x) - Integral (sec(x)tan^2(x) dx)
Use the identity tan^2(x) = sec^2(x) - 1.
Integral (sec^3(x) dx) = sec(x)tan(x) - Integral (sec(x)[sec^2(x)
- 1] dx)
Distribute the sec(x).
Integral (sec^3(x) dx) = sec(x)tan(x) - Integral( (sec^3(x) - sec(x)) dx)
Now, separate into two integrals.
Integral (sec^3(x) dx) = sec(x)tan(x) - [Integral (sec^3(x) dx) -
Integral (sec(x) dx)]
Distribute the minus over the brackets.
Integral (sec^3(x) dx) = sec(x)tan(x) - Integral (sec^3(x) dx) +
Integral (sec(x) dx)
Here's the part which gets tricky; we're going to move
- Integral (sec^3(x) dx) to the left hand side of our equation, resulting in TWO of them.
2 Integral (sec^3(x) dx) = sec(x)tan(x) + Integral (sec(x) dx)
And we know what the integral of sec(x) is (we stated it above).
2 Integral (sec^3(x) dx) = sec(x)tan(x) + ln|sec(x) + tan(x)|
All we have to do now is divide everything by 2, which is the same as multiplying everything by (1/2).
Integral (sec^3(x) dx) = (1/2)sec(x)tan(x) + (1/2)ln|sec(x) + tan(x)|
And don't forget to add the constant.
= (1/2)sec(x)tan(x) + (1/2)ln|sec(x) + tan(x)| + C
hope this helps u !! :D