calculate-A-dt-sin-t-by-using-u-cos-t-

Question Number 169249 by mathocean1 last updated on 26/Apr/22

c a l c u l a t e A = \int \frac{d t}{s i n (t)} b y u s i n g

u = c o s (t)

Commented by infinityaction last updated on 26/Apr/22

A = \int \frac{\sin t}{\sin^{2} t} d t

A = \int \frac{\sin t}{1 - \cos^{2} t} d t

u = \cos t \Rightarrow - d u = \sin t d t

A = - \int \frac{d u}{1 - u^{2}} \Rightarrow - \int \frac{d u}{(1 - u) (1 + u)}

A = \frac{- 1}{2} {\int \frac{d u}{1 - u} + \int \frac{d u}{1 + u}}

A = \frac{- 1}{2} {- \log (1 - u) + \log (1 + u)} + c

A = \frac{1}{2} \log ∣ \frac{1 - u}{1 + u} ∣ + c

A = \frac{1}{2} \log ∣ \frac{1 - \cos t}{1 + \cos t} ∣ + c

A = \frac{1}{2} \log ∣ \tan^{2} \frac{t}{2} ∣ + c

A = \log ∣ \tan \frac{t}{2} ∣ + c