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Q12  Find the integrals of the functions \frac{\sin ^ 2x }{1+ \cos x }

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Using trigonometric identities we can write the given integral as follows.

\frac{\sin ^ 2x }{1+ \cos x } 

\\=\frac{4\sin^2\frac{x}{2}\cos^2\frac{x}{2}}{2\cos^2\frac{x}{2}}\\ =2\sin^2\frac{x}{2}\\ =1-\cos x

\therefore \int \frac{sin^22x}{1+\cos x} = \int (1-\cos x)dx
                                 \\= \int 1dx-\int\cos x\ dx\\ =x-\sin x+C
 

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manish

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