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Integrals

JEE Main 2025 (Online) 2nd April Evening ShiftMCQMedium📈 Topic in 5% papers+4−1
Ques. 1JEE Main 2025, 2 April (Evening Shift)

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

  1. 2 - 2 - e(1 + 2)
  2. 2 + 2 + e(1 + 2)
  3. 2 + 2 - e(1 + 2)
  4. 2 - 2 + e(1 + 2)

Answer

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

Solution

Rationalise: 1 3 + x2 + 1 + x2 = 3 + x2 - 1 + x2(3 + x2) - (1 + x2) = 3 + x2 - 1 + x22

4 01 ( ) dx = 2 01 3 + x2 dx - 2 01 1 + x2 dx

Using x2 + a2 dx = x2 x2 + a2 + a22 e |x + x2 + a2 | + C:

01 3 + x2 dx = 12(2) + 32 e 3 - 32 e 3 = 1 + 34 e 3

01 1 + x2 dx = 22 + 12 e(1 + 2)

4 01 ( ) dx = 2 + 32 e 3 - 2 - e(1 + 2)

3 e 3 = 32 e 3

Required value = 2 - 2 - e(1 + 2)

Answer: (1)

Keep practising — Integrals

Ques. 2JEE Main 2025, 29 January (Morning Shift)

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

  1. 3 4
  2. 4 3
  3. 6 43
  4. 2 3
Ques. 3JEE 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. 4JEE Main 2025, 29 January (Evening Shift)

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

Ques. 5JEE Main 2025, 28 January (Morning Shift)

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

  1. 196
  2. 100
  3. 64
  4. 144
Ques. 6JEE 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:

  1. 13
  2. 11
  3. 9
  4. 15
Ques. 7JEE Main 2025, 28 January (Evening Shift)

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

  1. 4 ( e 2 - 2 )
  2. e2 2 + 2
  3. 2 - e2
  4. 4 ( e 2 + 2 )
Ques. 8JEE Main 2025, 28 January (Evening Shift)

Let f be a real valued continuous function defined on the positive real axis such that g(x) = 0x t f(t) dt. If g (x3 ) = x6 + x7, then value of r=115 f (r3 ) is :

  1. 270
  2. 340
  3. 310
  4. 320
Ques. 9JEE Main 2025, 24 January (Morning Shift)

If I(m,n)= 01xm-1(1-x)n-1dx, m,n>0, then I(9,14)+I(10,13) is

  1. I(9,1)
  2. I(1,13)
  3. I(19,27)
  4. I(9,13)
Ques. 10JEE Main 2025, 24 January (Evening Shift)

If 2x2+5x+9 x2+x+1 dx =x x2+x+1+ x2+x+1 + e |x+ 12+ x2+x+1 |+C, where C is the constant of integration, then +2 is equal to .

Ques. 11JEE Main 2025, 23 January (Morning Shift)

The value of e2e4 1x ( e (( ex)2+1 )-1e (( ex)2+1 )-1+e ((6- ex)2+1 )-1 )dx is

  1. 1
  2. loge 2
  3. e2
  4. 2

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