0-a-dx-a-sinx-a-cosx-a-R-test-for-a-1-

Question Number 76956 by behi83417@gmail.com last updated on 01/Jan/20

\underset{0}{\int^{a}} \frac{dx}{\sqrt{(a + sinx) (a + cosx)}} = ?

[a \in R^{+}, test for : a = 1]

Answered by john santu last updated on 02/Jan/20

f o r a = 1

l e t I = \int \frac{d x}{\sqrt{(1 + \sin x) (1 + \cos x)}}

= \int \frac{d x}{\sqrt{\frac{1 - \sin^{2} x}{1 - \sin x} (2 \cos^{2} (\frac{x}{2}))}}

= \int \frac{\sqrt{1 - \sin x}}{\sqrt{2} \cos x \cos (\frac{x}{2})} d x

l e t \tan (\frac{x}{2}) = u