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Question Number 68033 by mathmax by abdo last updated on 03/Sep/19

$$find\int \frac{dx}{1+sinx+sin\left(2x\right)}$$

Answered by MJS last updated on 04/Sep/19

$$\mathrm{I}\mathrm{think}\mathrm{I}\mathrm{did}\mathrm{this}\mathrm{before}$$$$\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{use}\mathrm{Weierstrass}$$$$\int \frac{dx}{1+\sin x+\sin 2x}=$$$$[t=\tan \frac{x}{2}\to dx=\frac{2dt}{t^2+1}]$$$$=2\int \frac{t^2+1}{(t+1)(t^3-3t^2+5t+1)}dt=$$$$=2\int \frac{t^2+1}{(t+1)(t-\alpha )(t-\beta )(t-\gamma )}dt$$$$\alpha =1+u+v$$$$\beta =1+(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i})u-(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i})v$$$$\gamma =1+(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i})u-(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i})v$$$$u=\sqrt[3]{-2+\frac{2}{9}\sqrt{87}}$$$$v=\sqrt[3]{2+\frac{2}{9}\sqrt{87}}$$$$\mathrm{now}\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{decompose}.\mathrm{the}\mathrm{path}\mathrm{is}\mathrm{easy}$$$$\mathrm{but}\mathrm{the}\mathrm{constants}\mathrm{are}\mathrm{terrible}$$

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