JEE Mains · Maths · STD 12 - 5. continuity and differentiation
If \(f(x) = \int_0^x {t(\sin \,\,x\, - \sin \,\,t)\,dt} \) then ?
- A \(f'''(x) + f'(x) = \cos \,x\, - 2x\,\,\sin \,x\)
- B \(f'''(x) + f''(x) - f'(x) = \cos \,x\,\)
- C \(f'''(x) - f''(x) = \cos \,x\,\, - \,2x\,\,\sin \,x\)
- D \(f'''(x) + f''(x) = \,\sin \,x\)
Answer & Solution
Correct Answer
(A) \(f'''(x) + f'(x) = \cos \,x\, - 2x\,\,\sin \,x\)
Step-by-step Solution
Detailed explanation
\(f\left( x \right) = \int_0^x {t\left( {\sin x - \sin t} \right)} .dt\) \( = \sin x\int_0^x {t.dt} - \int_0^x {t\sin t.dt} \) \( = \frac{{{x^2}}}{2}\sin x + \left[ {t\cos t_0^x} \right] + \sin x\) \( \Rightarrow f\left( x \right) = \frac{{{x^2}}}{2}\sin x + x\cos x + \sin x\)…
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