I'm trying to integrate the expression:?

Question

I'm trying to integrate the expression:

∫ -sin ²(t) / cos(t) dt

However, no matter how hard I bang my head against this problem, I just can't seem to solve it. I know the answer, which is -sec(t) + cos(t) (in this case, forget the C term), but I haven't a clue how to get there.

Answer 1

Integral [ -sin ²(t) / cos(t) dt ] = Integral [{cos^2(t)-1}/cos(t) dt]
= Integral [cos (t)-sec(t) dt]
= sin(t) - ln [sec t + tan t ]

where
Integral [sec (t)] = Integral[sec(t).{(sec t + tan t)/(sec t + tan t}]
= Integral[(sec^2(t) + sec t tan t )/(sec t + tan t}]

let u = sec t + tan t
du = sec t tan t + sec^2 t
Then
Integral[(sec^2(t) + sec t tan t )/(sec t + tan t}] = Integral (1/u)du
= ln u.

Answer 2

use trig identities... remember: sin^2(x)+cos^2(x) = 1? Obviously the other people gave the answer...I just thought I would remind you of the identity that brings you to the answer...

Answer 3

i assume you're familiar with matlab if no longer ask a pal or specify Q1,2,3 outline a symbolic variable say x use the combine function 'int' it is going to evaluate an indefinite imperative

Answer 4

Replace -(sin^2)(t) with (cos^2)(t)-1
when you divide each by cos(t) you will need to integrate
cost-1/cost which means you need to integrate cost - sect

Answer 5

∫ - sin ²(t) / cos(t) dt =
∫ (cos ²(t) - 1) / cos(t) dt =
∫ cos(t)dt - ∫ 1 / cos(t) dt =
- sin(t) + ln[sec(t) + tan(t)]