The number of elements in the set S = { x : x [ 0 , 100 ] and 0 x t 2 sin ( x - t ) ? d t = x 2 } is_____. [2026]

Question

The number of elements in the set

$S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} \sin (x - t) d t = x^{2}}$ is_____. [2026]

Smart Achievers · Accepted Answer

(16)

$\int_{0}^{x} t^{2} \sin (x - t) d t = x^{2}$

$Use by parts$

$- x^{2} \cos x + x^{2} + 2 x^{2} \cos x - 2 x \sin x$

$- x^{2} \cos x + 2 x \sin x + 2 \cos x - 2 = x$

$\cos x = 1$

$x = 0, 2 π, 4 π, \dots, 30 π$

$Total Elements = 16$

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