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Question Number 201224 by Calculusboy last updated on 02/Dec/23

$$\int \frac{1}{\sqrt{1+\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}\mathit{d}\mathit{x}$$

Answered by witcher3 last updated on 02/Dec/23

$$\sqrt{1+\sin \left(\mathrm{x}\right)}=\mathrm{hint}:\mid \sin \left(\frac{\mathrm{x}}{2}\right)+\cos \left(\frac{\mathrm{x}}{2}\right)\mid$$

Answered by Sutrisno last updated on 04/Dec/23

$$=\int \frac{1}{\sqrt{{sin}^2\frac{1}{2}x+{cos}^2\frac{1}{2}x+2sin\frac{1}{2}xcos\frac{1}{2}x}}dx$$$$=\int \frac{1}{\sqrt{{(sin\frac{1}{2}x+cos\frac{1}{2}x)}^2}}dx$$$$=\int \frac{1}{sin\frac{1}{2}x+cos\frac{1}{2}x}dx$$$$=\int \frac{1}{\sqrt{2}cos(\frac{1}{2}x-\frac{\pi }{4})}dx$$$$=\int \frac{sec(\frac{1}{2}x-\frac{\pi }{4})}{\sqrt{2}}dx$$$$=\frac{2}{\sqrt{2}}ln\mid sec(\frac{1}{2}x-\frac{\pi }{4})+tan(\frac{1}{2}x-\frac{\pi }{4})\mid +c$$

Commented by Calculusboy last updated on 04/Dec/23

$$\mathit{t}\mathit{h}\mathit{a}\mathit{n}\mathit{k}\mathit{s}\mathit{s}\mathit{i}\mathit{r}$$

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