QJEEM18Ma3a2c: JEE Main 2018 Mathematics - Class 12 JEEM18Ma3a2c Solution: If f ( x ) = ∫ 0 x t ( sin ⁡ x − sin ⁡ t ) d t then :

Question

Verified Solution • 2025–26 Session moderate

If

f (x) = \int_{0}^{x} t (\sin x - \sin t) d t

then :

Accepted Answer

f(x) =

\int_{0}^{x} t (\sin x - \sin t) . d t

= sin x

\int_{0}^{x} t . d t - \int_{0}^{x} t \sin t . d t

=

\frac{x^{2}}{2}

sin x +

{[t \cos t]}_{0}^{x}

+ sin x

\Rightarrow

f(x) =

\frac{x^{2}}{2}

sinx + xcosx + sinx

f'(x) =

\frac{x^{2}}{2}

cosx + 2cos x

f''(x) = x cos x

-

\frac{x^{2}}{2}

sin x

-

2sin x

f'''(x) = cos x

-

2x sin x

-

\frac{x^{2}}{2}

cos x

-

2cos x

∴

f'''(x) + f'(x) = cos x

-

2x sin x

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