The number of elements in the set $S=\left\{x: x \in[0,100]\text{ and }\int_{0}^{x} t^{2} \sin (x-t) d t=x^{2}\right\}$ is

JEE Main · 2026 · 23 Jan 2026 · Shift 2 · Definite Integrals

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}}

Answer: 16

Expanding

sin (x - t) = sin x cos t - cos x sin t

and integrating by parts twice,

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

. The equation

x^{2} - 2 + 2 cos x = x^{2}

reduces to

cos x = 1

, so

x = 2 k π

. Within

[0, 100]

the admissible values are

x = 0, 2 π, 4 π, \dots, 30 π

, since

30 π \approx 94.25 < 100 < 32 π

. These are

16

values, so the set

S

has

16

elements.

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