Mathematics

Evaluate the following integral:
$$\displaystyle \int { \cfrac { \sin { x }  }{ \sqrt { 4\cos ^{ 2 }{ x } -1 }  }  } dx$$


SOLUTION
$$I = \displaystyle \int \dfrac{\sin x}{\sqrt{4 \cos^2 x - 1}} dx$$

Let $$2 \cos x = t$$

$$\dfrac{dt}{dx} = 2 (-\sin x)$$

$$ \dfrac{-dt}{2} = \sin x dx$$

$$I = -\dfrac{1}{2} \displaystyle \int \dfrac{dt}{\sqrt{t^2 - 1^2}}$$

$$I = -\dfrac{1}{2} [\ln [t + \sqrt{t^2 - 1}] + c$$

put the value of t, we get

$$I = -\dfrac{1}{2} \ln [2 \cos x + \sqrt{(2 \cos x)^2 -1} ] + c$$

$$I = -\dfrac{1}{2} \ln [2 \cos x + \sqrt{4 \cos^2 x - 1} ] + c$$
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Subjective Medium Published on 17th 09, 2020
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