Definite Integration
Convolution Integral — Counting Solutions
nta_pyq_2026_jan
Grade 12
Question:
The number of elements in the set $S=\left\{x:x\in[0,100]\text{ and }\displaystyle\int_0^x t^2\sin(x-t)\,dt=x^2\right\}$ is _____
Step-by-Step Solution
Key Concept: Use convolution: $\int_0^x t^2\sin(x-t)\,dt=x^2*(\sin x)$. By substitution $u=x-t$: $=\int_0^x(x-u)^2\sin u\,du$. Expanding and integrating: $=x^2-2x\cos x+2\sin x-2$... full result: $=x^2+2\cos x-2$.
$x=2n\pi$, $n=0,1,\ldots,15$. 16 elements.
Correct Answer: 16