The value of $\smallint \,\,\frac{{\sec \,\,x\,\,dx}}{{\sqrt {\sin \,\,\left( {2x + {\rm{\theta }}} \right)\,\, + \,\,\sin \,\,{\rm{\theta }}} }}$ is

$\sqrt {2\,\,(\tan \,\,x\,\, + \,\,\tan \,\,{\rm{\theta }})\,\,\sec \,\,{\rm{\theta }}} + c$

$\sqrt {(\tan \,\,x\,\, + \,\,\tan \,\,{\rm{\theta }})\,\,\sec \,\,{\rm{\theta }}} + c$

$\sqrt {2\,\,(\sin \,\,\,x\,\, + \,\,\tan \,\,{\rm{\theta }})\,\,\sec \,\,{\rm{\theta }}} + c$

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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The value of

\int \,\,\frac{{\sec \,\,x\,\,dx}}{{\sqrt {\sin \,\,\left( {2x + \mathrm{\theta }} \right)\,\, + \,\,\sin \,\,\mathrm{\theta }} }}

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Correct option: B

\sqrt {2\,\,(\tan \,\,x\,\, + \,\,\tan \,\,\mathrm{\theta })\,\,\sec \,\,\mathrm{\theta }} + c

Option B (

\sqrt {2\,\,(\tan \,\,x\,\, + \,\,\tan \,\,\mathrm{\theta })\,\,\sec \,\,\mathrm{\theta }} + c

) is correct because it is the result of correctly evaluating the integral after trigonometric simplification and substitution.