JEE Main 2026 (23 January Shift 2)

LEVELJEE Advanced

Animated Solution for Mathematics - Definite Integration: The number of elements in the set $S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} sin (x - t) d t = x^{2}}$ is .........

Q: Why do we include $n=0$ in our count?

The problem defines the set $S$ for $x \in [0, 100]$. At $x=0$, the integral $\int_{0}^{0} t^2 \sin(0-t) dt$ is clearly $0$, and the right side $x^2$ is also $0$. Since $0=0$, $x=0$ is a valid solution. Never ignore the boundaries!

Enter Numerical Value:

Visualized Solution

Analyzing the Integral Equation

Given set $S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} sin (x - t) d t = x^{2}}$
Objective: Solve the integral equation for $x$ and count solutions in $[0, 100]$ .
Let $I = \int_{0}^{x} t^{2} sin (x - t) d t$ .

Applying the Property $\int_{0}^{a} f (t) d t = \int_{0}^{a} f (a - t) d t$

The term $sin (x - t)$ is difficult to integrate directly.
Use the definite integral property: $\int_{0}^{a} f (t) d t = \int_{0}^{a} f (a - t) d t$
This is often called the "King's Rule".

Substituting $t \to x - u$

Let's apply the property by replacing $t$ with $x - u$ .
The limits remain $0$ to $x$ .
$I = \int_{0}^{x} (x - u)^{2} sin (x - (x - u)) d u$
Simplifies to: $I = \int_{0}^{x} (x - u)^{2} sin (u) d u$

Expanding the Integrand

Expand the squared term: $(x - u)^{2} = x^{2} - 2 xu + u^{2}$
Substitute back: $I = \int_{0}^{x} (x^{2} - 2 xu + u^{2}) sin (u) d u$
Distribute $sin (u)$ and split into three integrals:
$I = x^{2} \int_{0}^{x} sin (u) d u - 2 x \int_{0}^{x} u sin (u) d u + \int_{0}^{x} u^{2} sin (u) d u$

Evaluating the First Integral

First term: $I_{1} = x^{2} \int_{0}^{x} sin (u) d u$
The integral of $sin (u)$ is $- cos (u)$ .
Apply limits from $0$ to $x$ :
$I_{1} = x^{2} [- cos (u)]_{0}^{x} = x^{2} (- cos (x) - (- cos (0)))$
$I_{1} = x^{2} (1 - cos (x))$

Evaluating the Second Integral

Second term: $I_{2} = - 2 x \int_{0}^{x} u sin (u) d u$
Use Integration by Parts (ILATE rule): $\int u sin (u) d u = - u cos (u) + sin (u)$
Apply limits from $0$ to $x$ :
$I_{2} = - 2 x [- x cos (x) + sin (x) - (0 + 0)]$
$I_{2} = 2 x^{2} cos (x) - 2 x sin (x)$

Evaluating the Third Integral

Third term: $I_{3} = \int_{0}^{x} u^{2} sin (u) d u$
Use Integration by Parts twice:
$\int u^{2} sin (u) d u = - u^{2} cos (u) + 2 u sin (u) + 2 cos (u)$
Apply limits from $0$ to $x$ :
$I_{3} = [- x^{2} cos (x) + 2 x sin (x) + 2 cos (x)] - [0 + 0 + 2]$
$I_{3} = - x^{2} cos (x) + 2 x sin (x) + 2 cos (x) - 2$

Combining All Terms

Summing the parts: $I = I_{1} + I_{2} + I_{3}$
$I = (x^{2} - x^{2} cos (x)) + (2 x^{2} cos (x) - 2 x sin (x)) + (- x^{2} cos (x) + 2 x sin (x) + 2 cos (x) - 2)$
Notice the cancellations:
$- x^{2} cos (x) + 2 x^{2} cos (x) - x^{2} cos (x) = 0$
$- 2 x sin (x) + 2 x sin (x) = 0$
Simplified Integral: $I = x^{2} + 2 cos (x) - 2$

Equating to $x^{2}$

Return to the original equation: $I = x^{2}$
Substitute our simplified $I$ :
$x^{2} + 2 cos (x) - 2 = x^{2}$
Subtract $x^{2}$ from both sides:
$2 cos (x) - 2 = 0$
$cos (x) = 1$

Solving $cos (x) = 1$

We need to find values of $x$ where $cos (x) = 1$ .
From trigonometry, $cos (x) = 1$ at $x = 0, 2 π, 4 π, 6 π, \dots$
General solution: $x = 2 nπ$ , where $n$ is an integer ( $n \in Z$ ).

Applying the Domain Constraint

The problem restricts $x$ to the interval $[0, 100]$ .
So, $0 \leq 2 nπ \leq 100$ .
Since $x \geq 0$ , $n$ must be a non-negative integer ( $n \geq 0$ ).
Divide by $2 π$ : $n \leq \frac{100}{2 π}$
Using $π \approx 3.14159$ , $\frac{100}{2 π} \approx \frac{100}{6.283} \approx 15.915$

Counting the Elements in Set $S$

We have $0 \leq n \leq 15.915$ .
Since $n$ is an integer, the possible values are $n = 0, 1, 2, \dots, 15$ .
The corresponding $x$ values are $0, 2 π, 4 π, \dots, 30 π$ .
The value $n = 16$ gives $x = 32 π \approx 100.53$ , which is outside the interval.
Total number of elements = $15 - 0 + 1 = 16$ .

00:00 / 00:00

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

We are tasked with finding 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}}

At first glance, this integral appears complex. However, we can simplify the expression by utilizing the symmetry property of definite integrals.

The King's Property

We invoke the property

\int_{0}^{a} f (t) d t = \int_{0}^{a} f (a - t) d t

. By applying this to our integral with

a = x

, we substitute

t = x - u

The integral transforms into:

I = \int_{0}^{x} (x - u)^{2} sin (u) d u

This substitution isolates the variable

u

within the sine function, making the integration process significantly more manageable.

The Expansion and Integration

Next, we expand the term

(x - u)^{2}

into

x^{2} - 2 xu + u^{2}

. Distributing

sin (u)

across these terms, we obtain:

I = x^{2} \int_{0}^{x} sin (u) d u - 2 x \int_{0}^{x} u sin (u) d u + \int_{0}^{x} u^{2} sin (u) d u

Since

x

is treated as a constant relative to the variable of integration

u

, we evaluate these integrals using standard methods and integration by parts (ILATE rule).

The evaluation yields:

I = x^{2} (1 - cos (x)) - 2 x (sin (x) - x cos (x)) + (2 x sin (x) - (x^{2} - 2) cos (x) - 2)

The Grand Cancellation

Upon simplifying the expression, the terms involving

x^{2} cos (x)

and

x sin (x)

cancel out perfectly. We are left with the simplified result:

I = x^{2} + 2 cos (x) - 2

Equating this to the right-hand side of our original equation,

x^{2}

, we get:

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

This simplifies further to:

2 cos (x) - 2 = 0 \Rightarrow cos (x) = 1

The Final Count

We must solve

cos (x) = 1

for

x \in [0, 100]

. The general solution is

x = 2 nπ

for integers

n

Given the interval

[0, 100]

, we find the values of

n

such that:

0 \leq 2 nπ \leq 100 \Rightarrow 0 \leq n \leq \frac{100}{2 π} \approx 15.91

Thus,

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

. There are exactly 16 values of

x

that satisfy the condition.

Common Traps & FAQs

Why can't we just use Leibniz's Rule to differentiate both sides?

You absolutely could! Differentiating both sides with respect to

x

would give you a differential equation. However, the resulting equation would involve integrals of

cos (x - t)

, leading to a second-order differential equation. While valid, it's a longer path. The King's Property is the elegant shortcut.

Why do we include

n = 0

in our count?

The problem defines the set

S

for

x \in [0, 100]

. At

x = 0

, the integral

\int_{0}^{0} t^{2} sin (0 - t) d t

is clearly

0

, and the right side

x^{2}

is also

0

. Since

0 = 0

x = 0

is a valid solution. Never ignore the boundaries!

Is it a coincidence that all the complex terms cancelled out?

In mathematics, especially in JEE problems, 'coincidence' is rarely the answer. The structure of the problem is designed to reward the use of the King's Property. The cancellation is the universe's way of telling you that you chose the right path.

Animated Solution for Mathematics - Definite Integration: The number of elements in the set $S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} sin (x - t) d t = x^{2}}$ is .........

Enter Numerical Value:

Visualized Solution

Analyzing the Integral Equation

Applying the Property $\int_{0}^{a} f (t) d t = \int_{0}^{a} f (a - t) d t$

Substituting $t \to x - u$

Expanding the Integrand

Evaluating the First Integral

Evaluating the Second Integral

Evaluating the Third Integral

Combining All Terms

Equating to $x^{2}$

Solving $cos (x) = 1$

Applying the Domain Constraint

Counting the Elements in Set $S$

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

The King's Property

The Expansion and Integration

The Grand Cancellation

The Final Count

Similar Questions

$\int_{0}^{10 π} ∣ sin x ∣ d x$ is

Let $a_{n} = \int_{- 1}^{n} (1 + \frac{x}{2} + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \dots + \frac{x ^{n - 1}}{n}) d x$ for $n \in N$ . Then the sum of all the elements of the set ${n \in N : a_{n} \in (2, 30)}$ is

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If $f (t) = \int_{0}^{π} \frac{2 x d x}{1 - c o s ^{2} t s i n ^{2} x}, 0 < t < π$ , then the value of $\int_{0}^{π /2} \frac{π ^{2} d t}{f ( t )}$ equals

The integral $\frac{24}{π} \int_{0}^{2} \frac{( 2 - x ^{2} ) d x}{( 2 + x ^{2} ) 4 + x ^{4}}$ is equal to ______.

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Animated Solution for Mathematics - Definite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of elements in the set S={x:x∈[0,100] and ∫0x​t2sin(x−t)dt=x2} is .........

Enter Numerical Value:

Visualized Solution

Analyzing the Integral Equation

Applying the Property .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∫0a​f(t)dt=∫0a​f(a−t)dt

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t→x−u

Expanding the Integrand

Evaluating the First Integral

Evaluating the Second Integral

Evaluating the Third Integral

Combining All Terms

Equating to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x2

Solving .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } cos(x)=1

Applying the Domain Constraint

Counting the Elements in Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

The King's Property

The Expansion and Integration

The Grand Cancellation

The Final Count

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Animated Solution for Mathematics - Definite Integration: The number of elements in the set $S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} sin (x - t) d t = x^{2}}$ is .........

Applying the Property $\int_{0}^{a} f (t) d t = \int_{0}^{a} f (a - t) d t$

Substituting $t \to x - u$

Equating to $x^{2}$

Solving $cos (x) = 1$

Counting the Elements in Set $S$

$\int_{0}^{10 π} ∣ sin x ∣ d x$ is

Let $a_{n} = \int_{- 1}^{n} (1 + \frac{x}{2} + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \dots + \frac{x ^{n - 1}}{n}) d x$ for $n \in N$ . Then the sum of all the elements of the set ${n \in N : a_{n} \in (2, 30)}$ is

$\frac{120}{π ^{3}} \int_{0}^{π} \frac{x ^{2} s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ is equal to _______.

Let $\frac{d}{d x} F (x) = \frac{e ^{s i n x}}{x}, x > 0$ . If $\int_{1}^{4} \frac{2 e ^{s i n x^{2}}}{x} d x = F (k) - F (1)$ , then one of the possible values of $k$ is $\dots\dots\dots$

If $[x]$ is the greatest integer $\leq x$ , then $π^{2} \int_{0}^{2} (sin \frac{π x}{2}) (x - [x])^{[x]} d x$ is equal to :

The value of the integral $\int_{- 2}^{2} \frac{s i n ^{2} x}{[ x / π ] + 1/2} d x$ (where $[x]$ denotes the greatest integer less than or equal to $x$ ) is :

If $f (t) = \int_{0}^{π} \frac{2 x d x}{1 - c o s ^{2} t s i n ^{2} x}, 0 < t < π$ , then the value of $\int_{0}^{π /2} \frac{π ^{2} d t}{f ( t )}$ equals

The integral $\frac{24}{π} \int_{0}^{2} \frac{( 2 - x ^{2} ) d x}{( 2 + x ^{2} ) 4 + x ^{4}}$ is equal to ______.

If $\int_{0}^{π} x f (sin x) d x = A \int_{0}^{π /2} f (sin x) d x$ , then $A$ is

The value $9 \int_{0}^{9} [\frac{10 x}{x + 1}] d x$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is _____