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Question Number 142687 by iloveisrael last updated on 04/Jun/21

$$\int \frac{dx}{\sqrt{1-\sin x}\sqrt{1+\cos x}}=?$$

Answered by qaz last updated on 04/Jun/21

$$\int \frac{\mathrm{dx}}{\sqrt{1-\sin \mathrm{x}}\sqrt{1+\cos \mathrm{x}}}$$$$=\frac{1}{\sqrt{2}}\int \frac{\mathrm{dx}}{(\sin \frac{\mathrm{x}}{2}-\cos \frac{\mathrm{x}}{2})\cos \frac{\mathrm{x}}{2}}$$$$=\frac{1}{\sqrt{2}}\int \frac{\mathrm{dx}}{2\sin \frac{\mathrm{x}}{2}\cos \frac{\mathrm{x}}{2}-\cos {}^2\frac{\mathrm{x}}{2}}$$$$=\frac{1}{\sqrt{2}}\int \frac{\sec {}^2\frac{\mathrm{x}}{2}}{2\tan \frac{\mathrm{x}}{2}-1}\mathrm{dx}$$$$=\frac{1}{\sqrt{2}}\ln \mid 2\tan \frac{\mathrm{x}}{2}-1\mid +\mathrm{C}$$

Commented by iloveisrael last updated on 05/Jun/21

$$\sqrt{1-\sin x}=\mid \sin \frac{x}{2}-\cos \frac{x}{2}\mid$$

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