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Question Number 72153 by 20190927 last updated on 25/Oct/19

$$\int \frac{\sin {}^3\mathrm{x}}{\sqrt{\cos \mathrm{x}}}\mathrm{dx}$$$$\int \sin \left(\mathrm{lnx}\right)\mathrm{dx}$$

Commented by Prithwish sen last updated on 25/Oct/19

$$\mathbf{a})\mathrm{let}\mathrm{cosx}={\mathrm{t}}^2\Rightarrow -\mathrm{sinx}\mathrm{dx}=2\mathrm{tdt}$$$$-\int \frac{(1-{\mathrm{t}}^4).2\mathrm{tdt}}{\mathrm{t}}=-2[\int \mathrm{dt}-\int {\mathrm{t}}^4\mathrm{dt}]$$$$=-2[\mathrm{t}-\frac{{\mathrm{t}}^5}{5}]=2[\frac{1}{5}{\mathbf{cos}}^{\frac{5}{2}}\mathbf{x}-\sqrt{\mathbf{cosx}}]+\mathbf{C}$$$$\mathbf{b})\mathrm{let}\mathrm{x}={\mathrm{e}}^{\mathrm{t}}\Rightarrow \mathrm{dx}={\mathrm{e}}^{\mathrm{t}}\mathrm{dt}$$$$\int \mathrm{sint}.{\mathrm{e}}^{\mathrm{t}}\mathrm{dt}=\mathrm{I}\mathrm{applying}\mathrm{by}\mathrm{parts}$$$$\mathrm{I}={\mathrm{e}}^{\mathrm{t}}\mathrm{sint}-\int {\mathrm{e}}^{\mathrm{t}}\mathrm{costdt}={\mathrm{e}}^{\mathrm{t}}.\mathrm{sint}-\{{\mathrm{e}}^{\mathrm{t}}\mathrm{cost}+\int {\mathrm{e}}^{\mathrm{t}}\mathrm{sintdt}\}$$$$2\mathrm{I}={\mathrm{e}}^{\mathrm{t}}(\mathrm{sint}-\mathrm{cost})$$$$\mathbf{I}=\frac{1}{2}\mathbf{x}\{\mathbf{sin}\left(\mathbf{lnx}\right)-\mathbf{cos}\left(\mathbf{lnx}\right)\}+\mathbf{C}$$$$\mathbf{please}\mathbf{check}.$$

Commented by 20190927 last updated on 25/Oct/19

$$\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much}$$

Commented by peter frank last updated on 26/Oct/19

$$thankyou$$

Answered by malwaan last updated on 25/Oct/19

$$\int \frac{{\mathit{s}\mathit{i}\mathit{n}}^3\mathit{x}}{\sqrt{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}\mathit{d}\mathit{x}=\int \frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}\left({\mathit{s}\mathit{i}\mathit{n}}^2\mathit{x}\right)}{{\mathit{c}\mathit{o}\mathit{s}}^{\frac{1}{2}}\mathit{x}}\mathit{d}\mathit{x}$$$$=\int {\mathit{c}\mathit{o}\mathit{s}}^{-\frac{1}{2}}\mathit{x}(1-{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x})\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{d}\mathit{x}$$$$=-\int ({\mathit{c}\mathit{o}\mathit{s}}^{-\frac{1}{2}}\mathit{x}-{\mathit{c}\mathit{o}\mathit{s}}^{\frac{3}{2}}\mathit{x})(-\mathit{s}\mathit{i}\mathit{n}\mathit{x})\mathit{d}\mathit{x}$$$$=-(2{\mathit{c}\mathit{o}\mathit{s}}^{\frac{1}{2}}\mathit{x}-\frac{2}{5}{\mathit{c}\mathit{o}\mathit{s}}^{\frac{5}{2}}\mathit{x})+\mathit{c}$$$$=\frac{2}{5}\sqrt{{\mathit{c}\mathit{o}\mathit{s}}^5\mathit{x}}-2\sqrt{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}+\mathit{c}$$$$=\frac{2}{5}{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x}\sqrt{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}-2\sqrt{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}+\mathit{c}$$

Commented by 20190927 last updated on 25/Oct/19

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

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