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Integrals

JEE Main 2026 (Online) 23rd January Evening ShiftMCQMedium📈 Topic in 21% papers+4−1
Ques. 1JEE Main 2026, 23 January (Evening Shift)

Let I(x) = 3 dx(4x + 6) ( 4x2 + 8x + 3 ) and I(0) = 34 + 20. If I ( 12 ) = a 2b + c, where a, b, c N, (a, b) = 1, then a + b + c is equal to :

  1. 30
  2. 29
  3. 28
  4. 31

Answer

(4) 31

Solution

Here 4x2 + 8x + 3 = (2x + 1)(2x + 3) and 4x + 6 = 2(2x + 3).

Put t = 2x + 3, so dx = dt2 and 2x + 1 = t - 2:

I = 3 dt22t t(t - 2) = 34 dtt2 1 - 2t

Now put 1 - 2t = u2, so 2t2 dt = 2u du, i.e. dtt2 = u du:

I = 34 u duu = 34u + C

Hence I(x) = 34 2x + 12x + 3 + C and I(0) = 34 1 3 + C = 34 + C.

I(0) = 34 + 20, we get C = 20, so

I ( 12 ) = 34 24 + 20 = 34 22 + 20 = 3 28 + 20

Thus a = 3, b = 8, c = 20 (and (3, 8) = 1).

a + b + c = 3 + 8 + 20 = 31

Answer: (4)

Keep practising — Integrals

Ques. 2JEE 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

Ques. 3JEE 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. 4JEE 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. 5JEE 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. 6JEE 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. 7JEE Main 2026, 21 January (Morning Shift)

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

Ques. 8JEE 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. 9JEE 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. 10JEE 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. 11JEE 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

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