The number of elements in the set S = x: x [0, 100] and 0^x t^2 (x-t) dt = x^2 is

Given: I = 0^x t^2 (x-t) dt and I = x^2, with x [0,100]. Find: The number of elements in the set S. Use the substitution u = x-t. Then I = 0^x (x-u)^2 u du so I = 0^x (x^2 - 2xu + u^2) u du. Therefore, I = x^2 0^x u du - 2x 0^x u u du + 0^x u^2 u du. Now evaluate the parts shown in the solution: 0^x u du = 1 - x 0^x u u du = x - x x 0^x u^2 u du = -x^2 x + 2x x + 2 x - 2. Substituting these, I = x^2(1- x) - 2x( x - x x) + (-x^2 x + 2x x + 2 x - 2). Simplifying, I = x^2 + 2 x - 2. Set I = x^2: x^2 + 2 x - 2 = x^2 2 x = 2 x = 1. Hence, x = 2n for integers n. Since x [0,100], 0 2n 100 0 n 50 15.9. Thus, n 0,1,2,,15, which gives 16 values. Therefore, the number of elements in the set is 16.

The number of elements in the set S = x: x [0, 100] and 0^x t^2 (… — Solution | JEE Mathematics

NVAMediumJEE 2026Properties of Definite Integrals

The number of elements in the set $S = \{ x : x \in [0, 100] \text{ and } \int_0^x t^2 \sin(x-t) \, dt = x^2 \}$ is _____

Answer

Correct answer:16

Step-by-step solution

Standard Method

Given:

I = \int_0^x t^2 \sin(x-t) \, dt

and

I = x^2,

with $x \in [0,100]$ .

Find: The number of elements in the set $S$ .

Use the substitution $u = x-t$ . Then

I = \int_0^x (x-u)^2 \sin u \, du

I = \int_0^x (x^2 - 2xu + u^2) \sin u \, du.

Therefore,

I = x^2 \int_0^x \sin u \, du - 2x \int_0^x u \sin u \, du + \int_0^x u^2 \sin u \, du.

Now evaluate the parts shown in the solution:

\int_0^x \sin u \, du = 1 - \cos x

\int_0^x u \sin u \, du = \sin x - x\cos x

\int_0^x u^2 \sin u \, du = -x^2\cos x + 2x\sin x + 2\cos x - 2.

Substituting these,

I = x^2(1-\cos x) - 2x(\sin x - x\cos x) + (-x^2\cos x + 2x\sin x + 2\cos x - 2).

Simplifying,

I = x^2 + 2\cos x - 2.

Set $I = x^2$ :

x^2 + 2\cos x - 2 = x^2

2\cos x = 2

\cos x = 1.

Hence,

x = 2n\pi

for integers $n$ . Since $x \in [0,100]$ ,

0 \le 2n\pi \le 100

0 \le n \le \frac{50}{\pi} \approx 15.9.

Thus,

n \in \{0,1,2,\dots,15\},

which gives $16$ values.

Therefore, the number of elements in the set is $16$ .

Common mistakes

Taking the answer key value $2$ without checking the solution working is incorrect. The solution derivation gives $\cos x = 1$ and hence multiple values of $x$ in $[0,100]$ , so the count must be computed from the interval.
Missing the value $x = 0$ is a common mistake. Since $0 = 2 \cdot 0 \cdot \pi$ and the interval is closed, $0$ must be included.
Using $n < \frac{50}{\pi}$ but then stopping at $n=14$ is incorrect. Because $\frac{50}{\pi} \approx 15.9$ , all integers from $0$ through $15$ are allowed.

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