The number of elements in the set
\( S = \{ x : x \in [0,100], \int_{0}^{x} t^2 \sin(x-t)\,dt = x^2 \} \)
is ______.
🎥 Video solution / Text Solution of this question is given below:
$\int_0^x t^2 \sin(x - t),dt = x^2$
Use integration by parts
$-x^2\cos x + x^2 + 2x\cos x - 2x\sin x - x^2\cos x + 2x\sin x + 2\cos x - 2 = x$
$\Rightarrow \cos x = 1$
$\Rightarrow x = 0, 2\pi, 4\pi, \ldots, 30\pi$
Total elements $= 16$