Mathematics

Solve:
$$\int \frac{dx}{\sqrt{4-x^{2}}}$$


SOLUTION
$$\int \dfrac{dx}{\sqrt4-x^2}  \ \ Let \  x=2\sin \theta$$

$$\dfrac{dx}{d \theta}=2 \cos \theta$$

$$=\int \dfrac{dx}{\sqrt{4-4 \sin^2} \theta}$$

$$=\int \dfrac{2\cos \theta d\theta}{\sqrt{4-4 \sin^2} \theta}$$


$$=\int { \dfrac { 2\cos { \theta d\theta  }  }{ \sqrt { 4\left( 1-{ \sin  }^{ 2 }\theta  \right)  }  }  } $$

$$=\int { \dfrac { 2\cos { \theta d\theta  }  }{ \sqrt { 4\cos ^{ 2 }\theta  }  }  } $$

$$=\int { \dfrac { 2\cos { \theta d\theta  }  }{ 2\cos { \theta  }  }  } $$

$$=\int d\theta=\theta$$

$$\therefore \theta=\dfrac{1}{2}\sin^{-1}x+c.$$ Ans

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Subjective Medium Published on 17th 09, 2020
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