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JEE MATHEMATICS MCQ: DIFFERENTIATION

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Analyzing +4 | -1
If u=−f′′  (θ)  sin⁡  θ  +  f′  (θ)  cos⁡  θu = - f''\,\,\left( \mathrm{\theta } \right)\,\,\sin \,\,\mathrm{\theta }\,\, + \,\,f'\,\,\left( \mathrm{\theta } \right)\,\,\cos \,\,\mathrm{\theta } and v=f′′  (θ)  cos⁡  θ  +  f′  (θ)  sin⁡  θv = f''\,\,\left( \mathrm{\theta } \right)\,\,\cos \,\,\mathrm{\theta }\,\, + \,\,f'\,\,\left( \mathrm{\theta } \right)\,\,\sin \,\,\mathrm{\theta }, then

f  [(dudθ)2+  (dvdθ)2]12dθf\,\,{\left[ {{{\left( {\frac{{du}}{{d\mathrm{\theta }}}} \right)}^2} + \,\,{{\left( {\frac{{dv}}{{d\mathrm{\theta }}}} \right)}^2}} \right]^{\frac{1}{2}}}d\mathrm{\theta } is equal to
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Correct option: B
f  (θ)  +  f′′  (θ)  +  cf\,\,\left( \mathrm{\theta } \right)\,\, + \,\,f''\,\,\left( \mathrm{\theta } \right)\,\, + \,\,c
Option B (f (θ) + f′′ (θ) + cf\,\left( \mathrm{\theta } \right)\, + \,f''\,\left( \mathrm{\theta } \right)\, + \,c) is correct because after differentiating and simplifying, the integral evaluates to f(θ)+f′′(θ)+cf(\theta) + f''(\theta) + c.

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