Integrals

JEE Main 2025 (Online) 3rd April Morning ShiftMCQMedium📈 Topic in 66% papers+4−1

Ques. 1JEE Main 2025, 3 April (Morning Shift)

Let the domain of the function f(x) = 2 4 6 (3 + 4x - x2) be (a, b). If 0b-a [x2] dx = p - q - r, p, q, r N, (p, q, r) = 1, where [ ] is the greatest integer function, then p + q + r is equal to

Answer

(1) 10

Solution

Domain: 4 6 (3 + 4x - x2) > 0

⇒ 6 (3 + 4x - x2) > 1

⇒ 3 + 4x - x2 > 6 ⇒ x2 - 4x + 3 < 0 ⇒ 1 < x < 3

⇒ a = 1, b = 3, b - a = 2

02 [x2] dx = 01 0 dx + 1 2 1 dx + 2 3 2 dx + 32 3 dx

= ( 2 - 1) + 2( 3 - 2) + 3(2 - 3)

= 5 - 2 - 3

⇒ p = 5, q = 2, r = 3

∴ p + q + r = 10

Answer: (1)

Keep practising — Integrals

Ques. 2JEE Main 2025, 3 April (Evening Shift)

The integral 0 8x dx4 2 x + 2 x is equal to

2 2
4 2
2
3 22

Ques. 3JEE Main 2025, 2 April (Morning Shift)

Let [.] denote the greatest integer function. If 0e3 [ 1ex-1 ] dx = - e 2, then 3 is equal to .

Ques. 4JEE Main 2025, 2 April (Evening Shift)

Let (a, b) be the point of intersection of the curve x2 = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral I = ab 9x21 + 5x dx is equal to :

Ques. 5JEE Main 2025, 2 April (Evening Shift)

4 01 ( 1 3 + x2 + 1 + x2 )dx - 3 e( 3) is equal to :

2 - 2 - e(1 + 2)
2 + 2 + e(1 + 2)
2 + 2 - e(1 + 2)
2 - 2 + e(1 + 2)

Ques. 6JEE Main 2025, 29 January (Morning Shift)

The integral 80 0 4 ( + 9 + 16 2 )d is equal to :

3 4
4 3
6 43
2 3

Ques. 7JEE Main 2025, 29 January (Evening Shift)

If 24 0 4 ( |4x- 12 |+[2 x] )dx=2 + , where [ ] denotes the greatest integer function, then is equal to .

Ques. 8JEE Main 2025, 29 January (Evening Shift)

If t 0 ( 01(3x+5)t dx ) 1t= 5e ( 85 ) 23, then is equal to .

Ques. 9JEE Main 2025, 28 January (Morning Shift)

If - 2 2 96x2 2 x(1 + ex) dx = ( 2 + ), , Z, then ( + )2 equals

Ques. 10JEE Main 2025, 28 January (Evening Shift)

Let f: R R be a twice differentiable function such that f(2) = 1. If F(x) = x f(x) for all x R, 02 x F'(x) dx = 6 and 02 x2 F''(x) dx = 40, then F'(2) + 02 F(x) dx is equal to:

Ques. 11JEE Main 2025, 28 January (Evening Shift)

If f(x) = 1x1/4 (1 + x1/4 )dx, f(0) = -6, then f(1) is equal to :

4 ( e 2 - 2 )
e2 2 + 2
2 - e2
4 ( e 2 + 2 )

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