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
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