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  4. ∫ ex((1-sinx/1-cos x))dx is equal to

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Q. $\int e^{x}\left(\frac{1-sinx}{1-cos x}\right)dx $ is equal to

Integrals

Solution:

$\int e^{x}\left(\frac{1-sinx}{1-cos x}\right)dx = \int e^{x}\left(\frac{1-sinx}{2sin^{2} \frac{x}{2}}\right)dx $

$ =\int e^{x}\left(\frac{1}{2}cosec^{2} \frac{x}{2} -cot \frac{x}{2}\right) dx $

$ =\frac{1}{2} \int e^{x} cosec^{2} \frac{x}{2}dx - \int e^{x} cot \frac{x}{2} dx $

$= \frac{1}{2}\int e^{x} cosec^{2} \frac{x}{2}dx -e^{x}cot \frac{x}{2}-\frac{1}{2} \int e^{x} cosec^{2} \frac{x}{2}dx +C $

$ =-e^{x}cot \frac{x}{2}+C $