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JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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If ∫sin⁡  θ  −  cos⁡  θ(sin⁡  θ  +  cos⁡  θ)  sin⁡  θ  cos⁡  θ  +  sin⁡2  θ  cos⁡2  θdθ=cosec−1  (f(θ))+C\int \frac{{\sin \,\,\theta \,\, - \,\,\cos \,\,\theta }}{{\left( {\sin \,\,\theta \,\, + \,\,\cos \,\,\theta } \right)\,\,\sqrt {\sin \,\,\theta \,\,\cos \,\,\theta \,\, + \,\,{{\sin }^2}\,\,\theta \,\,{{\cos }^2}\,\,\theta } }}d\theta = \mathrm{cose}{\mathrm{c}^{ - 1}}\,\,\left( {f\left( \theta \right)} \right) + C, then
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Correct option: A
f(θ)=sin⁡  2θ  +  1f\left( \theta \right) = \sin \,\,2\theta \,\, + \,\,1
Option A (f(θ)=sin⁡ 2θ + 1f(\theta ) = \sin \,2\theta \, + \,1) is correct because after simplifying and integrating, the result matches the given form with this f(θ)f(\theta).

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