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Question Number 166192 by greogoury55 last updated on 15/Feb/22

$$\int \frac{dx}{5+4\sin x}=?$$

Answered by som(math1967) last updated on 15/Feb/22

$$\int \frac{dx}{5({sin}^2\frac{x}{2}+{cos}^2\frac{x}{2})+8sin\frac{x}{2}cos\frac{x}{2}}$$$$=\int \frac{{sec}^2\frac{x}{2}dx}{5{tan}^2\frac{x}{2}+5+8tan\frac{x}{2}}$$$$lettan\frac{x}{2}=z\Rightarrow {sec}^2\frac{x}{2}dx=2dz$$$$=2\int \frac{dz}{5z^2+8z+5}$$$$=\frac{2}{5}\int \frac{dz}{z^2+\frac{8z}{5}+1}$$$$=\frac{2}{5}\int \frac{dz}{z^2+2.z.\frac{4}{5}+{\left(\frac{4}{5}\right)}^2+1-{\left(\frac{4}{5}\right)}^2}$$$$=\frac{2}{5}\int \frac{dz}{{(z+\frac{4}{5})}^2+{\left(\frac{3}{5}\right)}^2}$$$$=\frac{2}{5}\times \frac{5}{3}\times {\tan }^{-1}\left(\frac{z+\frac{4}{5}}{\frac{3}{5}}\right)+C$$$$nowputz=tan\frac{x}{2}$$

Commented by peter frank last updated on 15/Feb/22

$$\mathrm{thank}\mathrm{you}$$

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