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Question Number 201519 by vahid last updated on 08/Dec/23

Answered by cortano12 last updated on 08/Dec/23

$$\left(1\right)\int \frac{\sin {}^3\mathrm{x}}{\cos {}^6\mathrm{x}}\mathrm{dx}=-\int \frac{1-\cos {}^2\mathrm{x}}{\cos {}^6\mathrm{x}}\mathrm{d}\left(\cos \mathrm{x}\right)$$

Answered by Calculusboy last updated on 08/Dec/23

$$2)\mathit{I}=\int \sqrt{\frac{1-\mathit{x}}{1+\mathit{x}}}\mathit{d}\mathit{x}\mathit{l}\mathit{e}\mathit{t}\mathit{x}=\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }\mathit{d}\mathit{x}=-2\mathit{s}\mathit{i}\mathit{n}2\mathit{\theta }\mathit{d}\mathit{\theta }$$$$\mathit{I}=\int \sqrt{\frac{1-\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }}{1+\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }}}-2\mathit{s}\mathit{i}\mathit{n}2\mathit{\theta }\mathit{d}\mathit{\theta }$$$$\mathit{N}\mathit{b}:1-\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }=2{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{\theta }\mathit{a}\mathit{n}\mathit{d}1+\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }=2{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{\theta }$$$$\mathit{I}=\int \sqrt{\frac{2{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{\theta }}{2{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{\theta }}}\cdot -2\mathit{s}\mathit{i}\mathit{n}2\mathit{\theta }\mathit{d}\mathit{\theta }[\mathit{N}\mathit{b}:\mathit{s}\mathit{i}\mathit{n}2\mathit{\theta }=2\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathit{a}\mathit{n}\mathit{d}{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{\theta }=\frac{1-\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }}{2}]$$$$\mathit{I}=-2\int \frac{\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }}{\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }}\cdot 2\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathit{d}\mathit{\theta }\iff \mathit{I}=-4\int {\mathit{s}\mathit{i}\mathit{n}}^2\mathit{\theta }\mathit{d}\mathit{\theta }$$$$\mathit{I}=-4\int \left(\frac{1-\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }}{2}\right)\mathit{d}\mathit{\theta }\iff \mathit{I}=-2\int (1-\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta })\mathit{d}\mathit{\theta }$$$$\mathit{I}=-2[\int 1\mathit{d}\mathit{\theta }-\int \mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }\mathit{d}\mathit{\theta }]$$$$\mathit{I}=-2[\mathit{\theta }-\frac{\mathit{s}\mathit{i}\mathit{n}2\mathit{\theta }}{2}]+\mathit{C}$$$$\mathit{I}=-2\mathit{\theta }+\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }+\mathit{C}\mathit{b}\mathit{u}\mathit{t}\mathit{\theta }=\frac{1}{2}{\mathit{c}\mathit{o}\mathit{s}}^{-1}\mathit{x}$$$$\mathit{I}={\mathit{c}\mathit{o}\mathit{s}}^{-1}\left(\mathit{x}\right)+\mathit{S}\mathit{i}\mathit{n}\left[\frac{1}{2}{\mathit{c}\mathit{o}\mathit{s}}^{-1}\left(\mathit{x}\right)\right]+\mathit{C}$$$$$$

Answered by Calculusboy last updated on 08/Dec/23

$$\mathit{I}=\int \frac{{\mathit{s}\mathit{i}\mathit{n}}^3\mathit{x}}{{\mathit{c}\mathit{o}\mathit{s}}^6\mathit{x}}\mathit{d}\mathit{x}\mathit{N}\mathit{b}:{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{x}=1-{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x}$$$$\mathit{I}=\int \frac{{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{x}\cdot \mathit{s}\mathit{i}\mathit{n}\mathit{x}}{{\mathit{c}\mathit{o}\mathit{s}}^6\mathit{x}}\mathit{d}\mathit{x}\mathit{l}\mathit{e}\mathit{t}\mathit{u}=\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathit{d}\mathit{u}=-\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{d}\mathit{x}$$$$\mathit{I}=-\int \frac{(1-{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x})\cdot \mathit{s}\mathit{i}\mathit{n}\mathit{x}}{{\mathit{u}}^6}\cdot \frac{\mathit{d}\mathit{u}}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}$$$$\mathit{I}=-\int \frac{(1-{\mathit{u}}^2)}{{\mathit{u}}^6}\mathit{d}\mathit{u}\iff \mathit{I}=-\int [\frac{1}{{\mathit{u}}^6}-\frac{{\mathit{u}}^2}{{\mathit{u}}^6}]\mathit{d}\mathit{u}$$$$\mathit{I}=-(\frac{{\mathit{u}}^{-5}}{-5}-\frac{{\mathit{u}}^{-3}}{-3})+\mathit{C}$$$$\mathit{I}=\frac{1}{5{\mathit{u}}^5}-\frac{1}{3{\mathit{u}}^3}+\mathit{C}\mathit{b}\mathit{u}\mathit{t}\mathit{u}=\mathit{c}\mathit{o}\mathit{s}\mathit{x}$$$$\mathit{I}=\frac{1}{5{\mathit{c}\mathit{o}\mathit{s}}^5\mathit{x}}-\frac{1}{3{\mathit{c}\mathit{o}\mathit{s}}^3\mathit{x}}+\mathit{C}$$

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