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Welcome to the Cypress College math review on 
integrating secant cubed

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we are going to integrate secant cubed

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we do this by using parts

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 here's the formula for using parts

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we are going to break secant cubed into secant 
times secant squared

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so u

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will be secant

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and dv

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 will be secant squared

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why do we do that

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because the antiderivative of secant squared is 
tangent yes

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you wouldn't want to put

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secant as dv because the antiderivative of secant 
is very messy

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yeah but the antiderivative of secant squared 
simplifies into tangent so it makes it much 
simpler

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right, du

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is simply

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secant times tangent

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right

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so from your formula we now have u times v so 
secant times tangent

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minus v du

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so we have

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 secant tangent and then another tangent

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so we have

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secant tangent squared

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now how are we going to integrate that

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we're gonna have to make the tangent squared 
into secants because u sub isn't going to work for 
that

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okay, we're going to need the Pythagorean identity 
for tangent squared, do you recall what that was, 

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well you remember the one for sines and 
cosines of course

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and if you divide that by cosine squared you are 
going to get the one for tangent squared

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so solve for tangent squared by subtracting the 
one, so secant squared minus one

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cool

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alright, so tangent squared is secant squared 
minus one

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and then we distribute

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oh no, look what showed up again

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the integral of secant cubed

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that's what we were trying to figure out

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what is he doing there again

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wait a minute

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we got one on the left side we got negative one 
on the right side

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what if we put those guys together what if we add 
the integral on the right side to the one on the left 
side and now we have two 

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of these integrals over on the left hand side 

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we put them together, we add like terms

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what do you think about that

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in a minute we will integrate secant

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I just do it at the end, divide by 2

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I just do it at the end, divide by 2

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and now we integrate secant

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do you recall what the antiderivative of secant 
was

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natural log, right

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natural log of the absolute value

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of secant theta

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plus tangent theta

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plus C, there we go

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now try it yourself

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integrate cosecant cubed

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and this is the answer that you should get