Mathematics

# $\displaystyle \int \dfrac{1}{\sqrt{1 - 4x^2}} dx$ on the limits $\left(-\dfrac{1}{2} , \dfrac{1}{2} \right)$

##### SOLUTION
$\int_{-\frac{1}{2}}^{\frac{1}{2}}\cfrac{1}{\sqrt{1-4x^{2}}}\, dx$

$=\int_{-\frac{1}{2}}^{\frac{1}{2}}\cfrac{1}{\sqrt{1-(2x)^{2}}}\, dx$

$=\left [ \cfrac{\sin^{-1}2x}{2} \right ]_{-\frac{1}{2}}^{\frac{1}{2}}$

$=\cfrac{1}{2}\left ( \sin^{-1}\left ( 2\times \cfrac{1}{2} \right )-\sin^{-1}\left (2\times \cfrac{-1}{2} \right ) \right )$

$=\cfrac{1}{2}\left ( \sin^{-1}1-\sin^{-1}(-1) \right )$

$=\cfrac{1}{2}\left ( \cfrac{\pi }{2}-\left ( -\cfrac{\pi }{2} \right ) \right )$

$=\cfrac{\pi }{2}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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