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Question Number 11741 by uni last updated on 30/Mar/17

$$\int {\mathrm{xtan}}^2\mathrm{x}\mathrm{dx}=?$$

Answered by mrW1 last updated on 30/Mar/17

$$\int x{\tan }^{2}xdx=\int {\tan }^{2}xd\left(\frac{x^2}{2}\right)$$$$=\frac{x^2{\tan }^{2}x}{2}-\int x^2\frac{\sin x}{{\cos }^{3}x}dx$$$$=\frac{x^2{\tan }^{2}x}{2}+\frac{1}{2}\int x^{2}d\left(\frac{1}{{\cos }^{2}x}\right)$$$$=\frac{x^2{\tan }^{2}x}{2}+\frac{x^2}{{2\cos }^{2}x}-\int \frac{x}{{\cos }^{2}x}dx$$$$=\frac{x^2(1+{\sin }^{2}x)}{{2\cos }^{2}x}-\int xd\left(\tan x\right)$$$$=\frac{x^2(1+{\sin }^{2}x)}{{2\cos }^{2}x}-x\tan x+\int \tan xdx$$$$=\frac{x^2(1+{\sin }^{2}x)}{{2\cos }^{2}x}-x\tan x-\ln \mid \cos x\mid +C$$

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