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JEE Main 202623 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2026 · 23 January · Evening Shift · Mathematics · Definite Integration

The number of elements in the set S=\ x: x ∈[0,100]. and .∫_ 0 ^ x...

The number of elements in the
set S={x:x∈[0,100]and∫0xt2sin⁡(x−t)dt=x2}\mathrm{S}=\left\{x: x \in[0,100]\right. and \left.\int_{0}^{x} t^{2} \sin (x-t) \mathrm{d} t=x^{2}\right\} is ____\_\_\_\_.

Correct answer

16

Step-by-step solution

Using u=x−tu = x - t: ∫0xt2sin⁡(x−t) dt=∫0x(x−u)2sin⁡u du\int_0^x t^2 \sin(x-t)\,dt = \int_0^x (x-u)^2 \sin u\,du.
Expanding and integrating term by term:
=x2(1−cos⁡x)−2x(−xcos⁡x+sin⁡x)+(−x2cos⁡x+2xsin⁡x+2cos⁡x−2)= x^2(1-\cos x) - 2x(-x\cos x + \sin x) + (-x^2\cos x + 2x\sin x + 2\cos x - 2)
=x2+2cos⁡x−2= x^2 + 2\cos x - 2.
Setting equal to x2x^2: 2cos⁡x−2=0⇒cos⁡x=1⇒x=2nπ2\cos x - 2 = 0 \Rightarrow \cos x = 1 \Rightarrow x = 2n\pi.
In [0,100][0, 100]: 2nπ≤100⇒n≤50π≈15.92n\pi \le 100 \Rightarrow n \le \frac{50}{\pi} \approx 15.9.
So n=0,1,2,…,15n = 0, 1, 2, \ldots, 15, giving 1616 elements.
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