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Question Number 61719 by aliesam last updated on 06/Jun/19

$$\int \frac{dx}{2+sin\left(x\right)}$$

Commented by maxmathsup by imad last updated on 07/Jun/19

$$letA=\int \frac{dx}{2+sinx}\Rightarrow A{=}_{tan\left(\frac{x}{2}\right)=t}\int \frac{1}{2+\frac{2t}{1+t^2}}\frac{2dt}{1+t^2}$$$$=2\int \frac{dt}{2+2t^2+2t}=\int \frac{dt}{t^2+t+1}=\int \frac{dt}{{(t+\frac{1}{2})}^2+\frac{3}{4}}{=}_{t+\frac{1}{2}=\frac{\sqrt{3}}{2}u}$$$$=\frac{4}{3}\int \frac{1}{1+u^2}\frac{\sqrt{3}}{2}du=\frac{2}{\sqrt{3}}arctan\left(u\right)+c=\frac{2}{\sqrt{3}}arctan\left(\frac{2t+1}{\sqrt{3}}\right)+c$$$$=\frac{2}{\sqrt{3}}arctan(\frac{2}{\sqrt{3}}tan\left(\frac{x}{2}\right)+\frac{1}{\sqrt{3}})+c.$$

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