If $\smallint \,\,\frac{{{e^x}(1 + \sin \,\,x)\,\,dx}}{{1 + \cos \,\,x}} = {e^x}f\,\left( x \right)\,\, + \,\,c,\;\,{\rm{then}}\;\,\,f\left( x \right)$ is equal to

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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Application +4 | -1

\int \,\,\frac{{{e^x}(1 + \sin \,\,x)\,\,dx}}{{1 + \cos \,\,x}} = {e^x}f\,\left( x \right)\,\, + \,\,c,\;\,\mathrm{then}\;\,\,f\left( x \right)

is equal to

View answer and explanation

Correct option: C

\tan \,\,\frac{x}{2}

Option C (

\tan \frac{x}{2}

) is correct because it is the function that satisfies the given integral equation after simplification and integration.