MCQ
જો $f(x) = \int_0^x {t(\sin \,\,x\, - \sin \,\,t)\,dt} $ તો ?
- 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$f'''(x) + f''(x) = \,\sin \,x$
$ = \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$
$f'\left( x \right) = \frac{{{x^2}}}{2}\cos x + 2\,\cos x$
$f''\left( x \right) = x\cos x - \frac{{{x^2}}}{2}\sin x - 2\sin x$
$f'''\left( x \right) = \cos x - 2x\sin x - \frac{{{x^2}}}{2}\cos x - 2\cos x$
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