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Question Number 163580 by mkam last updated on 08/Jan/22

$$\int {sin}^{2021}\left(x\right).sin\left(2023x\right)dx$$

Commented by mkam last updated on 08/Jan/22

$$??????$$

Answered by abdullahhhhh last updated on 08/Jan/22

$$\mathbf{I}=\int \mathbf{sin}(2022\mathbf{x}+\mathbf{x}){\mathbf{sin}}^{2021}\mathbf{x}\mathbf{dx}$$$$\int {\mathit{s}\mathit{i}\mathit{n}}^{2021}\mathit{x}[\mathit{s}\mathit{i}\mathit{n}\left(2022\mathit{x}\right)\mathit{c}\mathit{o}\mathit{s}\mathit{x}\left(\mathit{x}\right)+\mathit{c}\mathit{o}\mathit{s}\mathit{x}\left(2022\mathit{x}\right)\mathit{s}\mathit{i}\mathit{n}\mathit{x}]$$$$\mathbf{I}=\int \left[\mathbf{cosx}{\mathbf{sin}}^{2021}\mathbf{x}\mathbf{sin}\left(2022\mathbf{x}\right)\right]\mathbf{dx}+\int {\mathbf{sin}}^{2022}\mathbf{x}\mathbf{cos}\left(2022\mathbf{x}\right)\mathbf{dx}$$$$\mathbf{I}={\mathbf{I}}_1\left(\mathbf{By}\mathbf{parts}\right)+{\mathit{I}}_2$$$${\mathbf{I}}_1=\int \mathbf{cosx}{\mathbf{sin}}^{2021}\mathbf{x}\mathbf{sin}\left(2022\mathbf{x}\right)\mathbf{dx}$$$$\mathbf{u}=\mathbf{sin}\left(2022\mathbf{x}\right)\mathbf{dv}=\mathbf{cosx}{\mathbf{sin}}^{2021}\mathbf{x}$$$$\mathbf{du}=2022\mathbf{cos}\left(2022\mathbf{x}\right)\mathbf{v}=\frac{{\mathbf{sin}}^{2022}\mathbf{x}}{2022}$$$$\frac{1}{2022}\mathbf{sin}\left(2022\mathbf{x}\right){\mathbf{sin}}^{2022}\mathbf{x}-\int \mathbf{cos}\left(2022\right){\mathbf{sin}}^{2022}\mathbf{x}\mathbf{dx}$$$${\mathbf{I}}_1=\frac{\mathbf{sin}\left(2022\mathbf{x}\right){\mathbf{sin}}^{2022}\mathbf{x}}{2022}-{\mathbf{I}}_2$$$$\mathbf{I}=\frac{\mathbf{sin}\left(2022\mathbf{x}\right){\mathbf{sin}}^{2022}\mathbf{x}}{2022}+\mathbf{C}$$

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