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Question Number 142917 by ERA last updated on 07/Jun/21

$$\int \left({\mathbf{sin}}^7\left(\mathbf{x}\right)\right)\mathbf{dx}$$

Answered by Ar Brandon last updated on 07/Jun/21

$$2\mathrm{isin}\left(\mathrm{nx}\right)={\mathrm{z}}^{\mathrm{n}}-\frac{1}{{\mathrm{z}}^{\mathrm{n}}},2\mathrm{isinx}=\mathrm{z}-\frac{1}{\mathrm{z}}$$$$-{128\mathrm{isin}}^7\mathrm{x}={\mathrm{z}}^7-\frac{1}{{\mathrm{z}}^7}+7({\mathrm{z}}^5-\frac{1}{{\mathrm{z}}^5})+21({\mathrm{z}}^3-\frac{1}{{\mathrm{z}}^3})+35(\mathrm{z}-\frac{1}{\mathrm{z}})$$$$-{128\sin }^7\mathrm{x}=2\sin 7\mathrm{x}+14\sin 5\mathrm{x}+42\sin 3\mathrm{x}+70\mathrm{sinx}$$$$\int {\sin }^7\mathrm{xdx}=\frac{1}{128}(\frac{2\cos 7\mathrm{x}}{7}+\frac{14\cos 5\mathrm{x}}{5}+\frac{42\cos 3\mathrm{x}}{3}+70\mathrm{cosx})+\mathrm{C}$$

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