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Question

sinx1 cos x12.

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Solution

The given function is sin 2 x 1+cosx .

sin 2 x 1+cosx dx (1)

Also, sinx=2sin x 2 cos x 2 and 1+cosx=2 cos 2 x 2 (2)

From (1) and (2),

sin 2 x 1+cosx = ( 2sin x 2 cos x 2 ) 2 2 cos 2 x 2 =2 sin 2 x 2 =1cosx (3)

Put (3) in (1), we get,

sin 2 x 1+cosx dx = ( 1cosx )dx = 1dx cosxdx =xsinx+c

Thus, the integral of the function sin 2 x 1+cosx is xsinx+c.


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