Integrals

JEE Main 2026 (Online) 22nd January Evening ShiftNumericalHard📈 Topic in 67% papers+4

Ques. 1JEE 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 .

Answer

Solution

064 x1/3dx = 34 [x4/3 ]064 = 34 256 = 192

[x1/3] = k for k3 x < (k+1)3:

064 [x1/3]dx = 0(1) + 1(8 - 1) + 2(27 - 8) + 3(64 - 27)

= 7 + 38 + 111 = 156

= 192 - 156 = 36

Let f( ) = 2 6 + 6 ; f has period , so 036 f = 36 0 f

0 f = 0 /2 2 + 2 6 + 6 d (putting + 2 in the second half)

6 + 6 = 1 - 3 2 2 = 1 - 34 2 2

0 /2 d 1 - 34 2 2 = 0 /2 d 2 + 14 2 ( = 2 )

= 0 /2 4 2 d 4 + 2 = [2 -1 2 ]0 /2 =

1 036 f( ) d = 1 36 = 36

Answer: 36

Keep practising — Integrals

Ques. 2JEE Main 2026, 21 January (Morning Shift)

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

Ques. 3JEE Main 2026, 21 January (Morning Shift)

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

Ques. 4JEE 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. 5JEE 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. 6JEE Main 2025, 8 April (Evening Shift)

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

2 + 3
4 +
1 + 3
3 + 2

Ques. 7JEE 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

Ques. 8JEE Main 2025, 7 April (Morning Shift)

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

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

Ques. 9JEE Main 2025, 7 April (Evening Shift)

If ( 1x + 1x3 ) ( [23]3x-24 + x-26 ) dx = - 3( + 1) (3x + x ) + 1 + C, x > 0, ( , , Z), where C is the constant of integration, then + + is equal to .

Ques. 10JEE Main 2025, 4 April (Morning Shift)

The value of -11 (1 + |x| - x ) ex + ( |x| - x ) e-xex + e-x dx is equal to

1 + 2 23
1 - 2 23
2 + 2 23
3 - 2 23

Ques. 11JEE Main 2025, 4 April (Evening Shift)

Let f(x)+2f ( 1x )=x2+5 and 2g(x)-3g ( 12 )=x, x>0. If = 12 f(x) dx, and = 12 g(x) dx, then the value of 9 + is :

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