Welcome to the Cypress College math review on integrating secant cubed
we are going to integrate secant cubed
we do this by using parts
here's the formula for using parts
we are going to break secant cubed into secant times secant squared
so u
will be secant
and dv
will be secant squared
why do we do that
because the antiderivative of secant squared is tangent yes
you wouldn't want to put
secant as dv because the antiderivative of secant is very messy
yeah but the antiderivative of secant squared simplifies into tangent so it makes it much simpler
right, du
is simply
secant times tangent
right
so from your formula we now have u times v so secant times tangent
minus v du
so we have
secant tangent and then another tangent
so we have
secant tangent squared
now how are we going to integrate that
we're gonna have to make the tangent squared into secants because u sub isn't going to work for that
okay, we're going to need the Pythagorean identity for tangent squared, do you recall what that was,
well you remember the one for sines and cosines of course
and if you divide that by cosine squared you are going to get the one for tangent squared
so solve for tangent squared by subtracting the one, so secant squared minus one
cool
alright, so tangent squared is secant squared minus one
and then we distribute
oh no, look what showed up again
the integral of secant cubed
that's what we were trying to figure out
what is he doing there again
wait a minute
we got one on the left side we got negative one on the right side
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
of these integrals over on the left hand side
we put them together, we add like terms
what do you think about that
in a minute we will integrate secant
I just do it at the end, divide by 2
and now we integrate secant
do you recall what the antiderivative of secant was
natural log, right
natural log of the absolute value
of secant theta
plus tangent theta
plus C, there we go
now try it yourself
integrate cosecant cubed
and this is the answer that you should get