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Integrals

JEE Main 2026 (Online) 23rd January Evening ShiftNumericalMedium📈 Topic in 17% papers+4
Ques. 1JEE Main 2026, 23 January (Evening Shift)

The number of elements in the set S = \x : x [0, 100] and 0x t2 (x - t) dt = x2 \ is

Answer

16

Solution

Put x - t = u: 0x t2 (x - t) dt = 0x (x - u)2 u du

By parts: = [-(x - u)2 u ]0x - 0x 2(x - u) u du

= x2 - 2 0x (x - u) u du

Again by parts, 0x (x - u) u du = [(x - u) u ]0x + 0x u du = 1 - x, so

0x t2 (x - t) dt = x2 - 2 + 2 x

The given equation becomes x2 - 2 + 2 x = x2, so x = 1 and x = 2n with

0 2n 100 0 n 1002 15.9

so n = 0, 1, 2, , 15.

Number of elements = 16

Answer: 16

Keep practising — Integrals

Ques. 2JEE Main 2026, 22 January (Morning Shift)

The value of - 2 2 ( 1[x] + 4 ) dx, where [ ] denotes the greatest integer function, is

  1. 160(21 - 1)
  2. 160( - 7)
  3. 760( - 3)
  4. 760(3 - 1)
Ques. 3JEE Main 2026, 22 January (Morning Shift)

If ( x) -112( x) -52 dx = - p1q1( x) 92 - p2q2( x) 52 - p3q3( x) 12 + p4q4( x) -32 + C, where pi and qi are positive integers with (pi, qi) = 1 for i = 1, 2, 3, 4 and C is the constant of integration, then 15p1p2p3p4q1q2q3q4 is equal to

Ques. 4JEE Main 2026, 22 January (Evening Shift)

Let [.] be the greatest integer function. If = 064 (x1/3 - [x1/3] )dx, then 1 0 ( 2 6 + 6 )d is equal to .

Ques. 5JEE Main 2026, 21 January (Morning Shift)

The value of - /6 /6 ( + 4x111 - (|x| + /6) )dx is equal to:

  1. 8
  2. 4
  3. 2
  4. 6
Ques. 6JEE Main 2026, 21 January (Morning Shift)

6 0 |( 3x + 2x + x)| dx is equal to .

Ques. 7JEE Main 2026, 21 January (Evening Shift)

Let [ ] denote the greatest integer function and f(x) = n 1n3 k=1n [ k23x ]. Then 12 j=1 f(j) is equal to .

Ques. 8JEE Main 2026, 21 January (Evening Shift)

If 01 4 -1 (1 - 2x + 4x2 )dx = a -1(2) - b e(5), where a, b N, then (2a + b) is equal to .

Ques. 9JEE Main 2025, 8 April (Evening Shift)

The integral -1 32 ( | 2 x ( x) | )dx is equal to:

  1. 2 + 3
  2. 4 +
  3. 1 + 3
  4. 3 + 2
Ques. 10JEE Main 2025, 8 April (Evening Shift)

Let f(x) be a positive function and I1 = - 121 2x f(2x(1 - 2x)) dx and I2 = -12 f(x(1 - x)) dx. Then the value of I2I1 is equal to

  1. 12
  2. 9
  3. 6
  4. 4
Ques. 11JEE Main 2025, 7 April (Morning Shift)

The integral 0 (x+3) x1+3 2 x dx is equal to

  1. 3( +1)
  2. 3 3( +6)
  3. 3( +2)
  4. 2 3( +4)

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