Mathematics

Integrate : $$\displaystyle \int { \frac { 1+\tan { x }  }{ x+\log { \sec { x }  }  } dx } $$


SOLUTION
Given : $$\displaystyle \int { \frac { 1+\tan { x }  }{ x+\log { \sec { x }  }  } dx } $$

Let  $$I=\displaystyle \int { \frac { 1+\tan { x }  }{ x+\log { \sec { x }  }  } dx } $$

Let $$t=x+\log{\sec{x}}$$

$$\Rightarrow dt=\left(1+\dfrac{1}{\sec{x}}\sec{x}\tan{x}\right)dx$$

$$\Rightarrow dt=\left(1+\tan{x}\right)dx$$

$$\displaystyle \int{\dfrac{\left(1+\tan{x}\right)dx}{x+\log{\sec{x}}}}$$$$=\displaystyle \int{\dfrac{dt}{t}}$$

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

where $$(t=x+\log{\sec{x}})$$

$$I=\log{\left|x+\log{\sec{x}}\right|}+c$$ 

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Subjective Medium Published on 17th 09, 2020
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