Mathematics

Evaluate the following integrals:
$$\int { \sqrt { \cos { x } \sqrt { 4-\sin ^{ 2 }{ x }  }  }  } dx$$


SOLUTION
Let $$I=\displaystyle\int \cos x\sqrt{4-\sin^2 x}\ d x$$

$$\implies I=\displaystyle\int \sqrt{4-(\sin x)^2}\times \cos x d x$$

put $$\sin x=t\implies \cos x\ d x=d t$$

$$\implies I=\displaystyle\int \sqrt{2^2-t^2}\ d t$$

As we know that

$$\displaystyle\int \sqrt{a^2-x^2} d x=\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\text{sin}^{-1} \left(\dfrac{x}{a}\right)+C$$

Here $$a=2\implies a^2=4$$

$$\implies I=\dfrac{t}{2}\sqrt{4-t^2}+\dfrac{4}{2}\text{sin}^{-1} \left(\dfrac{t}{2}\right)+C$$

$$\implies I=\dfrac{1}{2}\sin x\sqrt{4-\sin^2 x}+2\text{sin}^{-1}\left(\dfrac{\sin x}{2}\right)+C$$
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Subjective Medium Published on 17th 09, 2020
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