Integral from -pi/3 to 0 of 3sin^3(x)dx ?

[Lo.Lee.Ta.]

∫from -\pi/3 to 0 of 3sec³xdx

Integration by parts...

3∫sec²x*secxdx

dv= 3sec²x
v= 3tanx

u= secx
du= secxtanx

Formula: uv - ∫vdu

secx*3tanx - ∫3tanx*secxtanxdx

Integration by parts again...

u= 3tanx
du= 3sec²x

dv= sextanx
v= secx

3tanx*secx - ∫secx*3sec²x

Altogether:

∫3sec³x = 3secxtanx - 3secxtanx - ∫3sec³x

*Add ∫3sec³x on both sides*

∫2*3sec³x = 0 !

It seems that the two 3secxtanx's would cancel! :/

∫6sec³x = 0

This is like the same thing we started out with! ugh, man. I don't really know what to do now.
I know this is a definite integral, so we should substitute -pi/3 and 0 in the end, but we still have an integral! #=_=

...This definitely does not seem right! :(
But I don't really see what's wrong with my work.
Please help! Thank you so much! :D

 

[Mark44]

∫from -\pi/3 to 0 of 3sec³xdx

Integration by parts...

3∫sec²x*secxdx

dv= 3sec²x
v= 3tanx

u= secx
du= secxtanx

Formula: uv - ∫vdu

secx*3tanx - ∫3tanx*secxtanxdx

Integration by parts again...

That's not the way to go. Your integral above can be written as
$\int tan^2(x) sec(x) dx$
$= \int (sec^2(x) - 1) sec(x) dx$
This will give you another sec³ integral and one that's somewhat easier. You should be able to solve algebraically for the integral you're looking for.
I am not including the factor of 3 from the first integral. It would have been better to bring it out of the integral right away and just concentrate on ∫sec³(x) dx.

u= 3tanx
du= 3sec²x

dv= sextanx
v= secx

3tanx*secx - ∫secx*3sec²x

Altogether:

∫3sec³x = 3secxtanx - 3secxtanx - ∫3sec³x

*Add ∫3sec³x on both sides*

∫2*3sec³x = 0 !

It seems that the two 3secxtanx's would cancel! :/

∫6sec³x = 0

This is like the same thing we started out with! ugh, man. I don't really know what to do now.
I know this is a definite integral, so we should substitute -pi/3 and 0 in the end, but we still have an integral! #=_=

...This definitely does not seem right! :(
But I don't really see what's wrong with my work.
Please help! Thank you so much! :D