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Question Number 132470 by physicstutes last updated on 14/Feb/21

$$\int {\tan }^{3}xdx$$

Answered by mindispower last updated on 14/Feb/21

$$\int tg\left(x\right)(1+{tg}^2\left(x\right))dx-\int tg\left(x\right)dx$$$$=\frac{{tg}^2\left(x\right)}{2}+ln\mid cos\left(x\right)\mid +c$$

Answered by bemath last updated on 14/Feb/21

$$\int \frac{(1-\cos {}^2\mathrm{x})\sin \mathrm{x}}{\cos {}^3\mathrm{x}}\mathrm{dx}$$$$\mathrm{let}\cos \mathrm{x}=\mu$$$$\int \frac{(1-{\mu }^2)(-\mathrm{d}\mu )}{{\mu }^3}=\int (\frac{1}{\mu }-{\mu }^{-3})\mathrm{d}\mu$$$$=\ln \mid \mu \mid +\frac{1}{2{\mu }^2}+\mathrm{c}$$$$=\ln \mid \cos \mathrm{x}\mid +\frac{1}{2}\sec {}^2\mathrm{x}+\mathrm{c}$$

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