Mathematics

Evaluate the following integral
$$\int { \cfrac { \sec { x } \tan { x }  }{ 3\sec { x } +5 }  } dx$$


SOLUTION
$$I=\displaystyle\int{\dfrac{\sec{x}\tan{x}}{3\sec{x}+5}dx}$$

Let $$t=3\sec{x}+5\Rightarrow\,dt=3\sec{x}\tan{x}\,dx$$

$$\Rightarrow\,\dfrac{dt}{3}=\sec{x}\tan{x}\,dx$$

$$I=\dfrac{1}{3}\displaystyle\int{\dfrac{dt}{t}}$$

$$I=\dfrac{1}{3}\log{\left|t\right|}+c$$

$$I=\dfrac{1}{3}\log{\left|3\sec{x}+5\right|}+c$$ where $$t=3\sec{x}+5$$
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Subjective Medium Published on 17th 09, 2020
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