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 Q 18   Integrate the functions e ^x \left ( \frac{1+ \sin x }{1+ \cos x } \right )

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Let 
I =e ^x \left ( \frac{1+ \sin x }{1+ \cos x } \right )
substitute 1 =\sin ^2\frac{x}{2}+\cos^2\frac{x}{2} and \sin x = 2\sin\frac{x}{2}\cos\frac{x}{2}

\\\Rightarrow e^x(\frac{\sin^2\frac{x}{2}+\cos^2\frac{x}{2}+2\sin\frac{x}{2}\cos\frac{x}{2}}{2\cos^2\frac{x}{2}})\\ =e^x(\frac{1}{2}\sec^2\frac{x}{2}+\tan\frac{x}{2})\\
let 
f(x) =\tan\frac{x}{2} \Rightarrow f'(x)=\frac{1}{2}\sec^2\frac{x}{2}
It is known that \int e^x[f(x)+f'(x)]=e^x[f(x)]+C
Therefore the solution of the given integral is 

I = e^x\tan\frac{x}{2} +C

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manish

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