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Question Number 155883 by john_santu last updated on 05/Oct/21

$$\int \frac{\tan {}^{2}x}{1-\tan {}^{2}x}dx=?$$

Commented by aliyn last updated on 05/Oct/21

$$\mathit{I}=\int \frac{{\mathit{t}\mathit{a}\mathit{n}}^2\mathit{x}}{1-{\mathit{t}\mathit{a}\mathit{n}}^2\mathit{x}}\mathit{d}\mathit{x}$$$$$$$$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:$$$$$$$$\mathit{I}=\int \frac{{\mathit{s}\mathit{e}\mathit{c}}^2\mathit{x}-1}{{\mathit{s}\mathit{e}\mathit{c}}^2\mathit{x}({\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x}-{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{x})}\mathit{d}\mathit{x}$$$$$$$$\mathit{I}=\int \frac{{\mathit{s}\mathit{e}\mathit{c}}^2\mathit{x}-1}{{\mathit{s}\mathit{e}\mathit{c}}^2\mathit{x}\left(\mathit{c}\mathit{o}\mathit{s}2\mathit{x}\right)}\mathit{d}\mathit{x}=\int \frac{\mathit{d}\mathit{x}}{\mathit{c}\mathit{o}\mathit{s}2\mathit{x}}-\int \frac{{\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x}}{\mathit{c}\mathit{o}\mathit{s}2\mathit{x}}\mathit{d}\mathit{x}$$$$$$$$\mathit{I}=\int \mathit{s}\mathit{e}\mathit{c}2\mathit{x}\mathit{d}\mathit{x}-\frac{1}{2}\int \frac{(1+\mathit{c}\mathit{o}\mathit{s}2\mathit{x})}{\mathit{c}\mathit{o}\mathit{s}2\mathit{x}}\mathit{d}\mathit{x}$$$$$$$$\mathit{I}=\frac{1}{2}\int \mathit{s}\mathit{e}\mathit{c}2\mathit{x}\times \frac{\mathit{s}\mathit{e}\mathit{c}2\mathit{x}+\mathit{t}\mathit{a}\mathit{n}2\mathit{x}}{\mathit{s}\mathit{e}\mathit{c}2\mathit{x}+\mathit{t}\mathit{a}\mathit{n}2\mathit{x}}\mathit{d}\mathit{x}-\frac{1}{2}\int \left(1\right)\mathit{d}\mathit{x}$$$$$$$$\mathit{I}=\frac{1}{4}\mathit{l}\mathit{n}\mid \mathit{s}\mathit{e}\mathit{c}2\mathit{x}+\mathit{t}\mathit{a}\mathit{n}2\mathit{x}\mid -\frac{1}{2}\mathit{x}+\mathit{C}$$$$$$$$◻\mathit{M}$$

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