Mathematics

Evaluate the following integral
$$\int { \cfrac { \sec ^{ 2 }{ x }  }{ \tan { x } +2 }  } dx$$


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

$$\displaystyle\int{\dfrac{{\sec}^{2}{x}}{\tan{x}+2}dx}$$

$$=\displaystyle\int{\dfrac{dt}{t}}$$

$$=\log{\left|t\right|}+c$$

$$=\log{\left|\tan{x}+2\right|}+c$$ where $$t=\tan{x}+2$$

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Subjective Medium Published on 17th 09, 2020
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