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Class-12-science»Maths

prove that integration of root(1+secx)dx = sin inverse {root2 sin(x/2)}

\int \sqrt{1 + secx} dx = \int \sqrt{\frac{1 + cosx}{cosx}} dx = \int \sqrt{\frac{(1 + cosx) (1 - cosx)}{cosx (1 - cosx)}} dx = \int \sqrt{\frac{1 - \cos^{2} x}{cosx (1 - cosx)}} dx = \int \frac{sinx}{\sqrt{cosx (1 - cosx)}} dx let cosx = t such that - sinxdx = dt = \int \frac{- dt}{\sqrt{t (1 - t)}} = \int \frac{- dt}{\sqrt{t - t^{2} + \frac{1}{4} - \frac{1}{4}}} = \int \frac{- dt}{\sqrt{{(\frac{1}{2})}^{2} - {(t - \frac{1}{2})}^{2}}} = - \sin^{- 1} \frac{(t - \frac{1}{2})}{\frac{1}{2}} = - \sin^{- 1} (2 cosx - 1) = \sin^{- 1} (1 - 2 cosx)

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