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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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