# Integral of sec^3(x) without reduction (1 Viewer)

#### jjjcjjj892

##### New Member
Can anyone help me integrate sec^3(x) without using the reduction formula?

#### Drongoski

##### Well-Known Member
do you allow integration by parts?

#### jjjcjjj892

##### New Member
Yea no problem, only issue I have is with just going straight to a reduction formula, I want to see how the answer progresses to the solution rather than going from a -> b.

#### Drongoski

##### Well-Known Member
I think you can do it this way with IBP:

$\bg_white I = \int sec^3 x dx = \int sec x dtanx = secx tanx - \int tanx dsecx \\ \\ = secxtanx - \int tanx \times secxtan x dx = secxtanx - \int secxtan^2 xdx \\ \\ = secxtanx - \int sec^3x dx + \int secx dx = secxtanx - I + \int secx dx \\ \\ = secxtanx - I + ln|secx + tan x | + 2C\implies 2I = \cdots$

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##### -insert title here-
Alternatively you can differentiate sec³x and then rearrange both sides and re-integrate. However this is essentially the same as Integration By parts.

"Inspection"

#### jjjcjjj892

##### New Member
I think you can do it this way with IBP:

$\bg_white I = \int sec^3 x dx = \int sec x dtanx = secx tanx - \int tanx dsecx \\ \\ = secxtanx - \int tanx \times secxtan x dx = secxtanx - \int secxtan^2 xdx \\ \\ = secxtanx - \int sec^3x dx + \int secx dx = secxtanx - I + \int secx dx \\ \\ = secxtanx - I + ln|secx + tan x | + 2C\implies 2I = \cdots$
Thanks for the reply, the way I progressed was going:

$\bg_white \\ \\I = \int sec^3x \ dx = \int 1/cos^3x \ dx = \int (cos^2x + sin^2x)/cos^3x \ dx \\ \\ = \int secx \ dx + \int sin^2x/cos^3x \ dx$

And its at this point I get lost, I can get the sec integral fine, its given, but the integral $\bg_white \int sin^2x/cos^3x$ is confusing

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#### Squar3root

##### listening to tapes
Thanks for the reply, the way I progressed was going:

$\bg_white \\ \\I = \int sec^3x \ dx = \int 1/cos^3x \ dx = \int (cos^2x + sin^2x)/cos^3x \ dx \\ \\ = \int secx \ dx + \int sin^2x/cos^3x \ dx$

And its at this point I get lost, I can get the sec integral fine, its given, but the integral $\bg_white \int sin^2x/cos^3x$ is confusing
use the identity tan^2(x) = sec^2(x) - 1

#### Squar3root

##### listening to tapes
that spoiler was a fail

#### jjjcjjj892

##### New Member
Thank you, that made it very clear cheers everyone for the help.