calculate-I-0-pi-2-dx-1-cosx-and-J-0-pi-2-cosx-1-cosx-dx-

Question Number 27757 by abdo imad last updated on 14/Jan/18

c a l c u l a t e I = \int_{0}^{\frac{π}{2}} \frac{d x}{1 + c o s x} a n d J = \int_{0^{}}^{\frac{π}{2}} \frac{c o s x}{1 + c o s x} d x .

Answered by mrW2 last updated on 14/Jan/18

\int \frac{d x}{1 + \cos x} = \int \frac{1}{\cos^{2} \frac{x}{2}} d (\frac{x}{2}) = \tan \frac{x}{2} + C

\int \frac{\cos x d x}{1 + \cos x} = \int \frac{1 + \cos x - 1}{1 + \cos x} d x = x - \int \frac{d x}{1 + \cos x} = x - \tan \frac{x}{2} + C

\Rightarrow I = {[\tan \frac{x}{2}]}_{0}^{\frac{π}{2}} = \tan \frac{π}{4} = 1

\Rightarrow J = {[x - \tan \frac{x}{2}]}_{0}^{\frac{π}{2}} = \frac{π}{2} - \tan \frac{π}{4} = \frac{π}{2} - 1

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