$\smallint {e^x}\,\,\left( {\frac{{1 - \,\,\sin \,\,x}}{{1 - \,\,\cos \,\,x}}} \right)\;dx$ is equal to

$ - \,{e^x}\,{\rm{cot}}\,\,\left( {\frac{x}{2}} \right) + c$

$ - \,{e^x}\,\tan \,\,\left( {\frac{x}{2}} \right) + c$

$ - \,\,\frac{1}{2}{e^x}\,\tan \,\,\left( {\frac{x}{2}} \right) + c$

$\frac{1}{2}\,\,{e^x}\,{\rm{cot}}\,\,\left( {\frac{x}{2}} \right) + c$

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

1 Medium

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Application +4 | -1

\int {e^x}\,\,\left( {\frac{{1 - \,\,\sin \,\,x}}{{1 - \,\,\cos \,\,x}}} \right)\;dx

is equal to

View answer and explanation

Correct option: B

- \,{e^x}\,\mathrm{cot}\,\,\left( {\frac{x}{2}} \right) + c

Option B (

\ - \,{e^x}\,\mathrm{cot}\,\,\left( {\frac{x}{2}} \right) + c

) is correct because after simplifying the integrand and applying integration by parts, the result matches this option.