Mathematics

Evaluate the following integrals:
$$\displaystyle  \int { \tan ^{ 3 }{ x } \sec ^{ 2 }{ x }  } dx$$


SOLUTION
$$\displaystyle\int{{\tan}^{3}{x}{\sec}^{2}{x}dx}$$

Let $$t=\tan{x}\Rightarrow\,dt={\sec}^{2}{x}dx$$

$$=\displaystyle\int{{t}^{3}\,dt}$$

We know that $$\displaystyle\int{{x}^{n}dx}=\dfrac{{x}^{n+1}}{n+1}+c$$

$$=\dfrac{{t}^{4}}{4}+c$$

$$=\dfrac{{\tan}^{4}{x}}{4}+c$$ where $$t=\tan{x}$$
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Subjective Medium Published on 17th 09, 2020
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