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Integrals

JEE Main 2025 (Online) 28th January Morning ShiftMCQMedium📈 Topic in 67% papers+4−1
Ques. 1JEE 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

Answer

(2) 100

Solution

Let I = - 2 2 96x2 2 x1 + ex dx. Using x -x,

I = - 2 2 96x2 2 x ex1 + ex dx

Adding the two forms,

2I = - 2 2 96x2 2 x dx = 2 0 2 96x2 2 x dx

I = 48 0 2 x2(1 + 2x) dx

Here 0 2 x2 dx = 324, and since x2 2x dx = x2 2x2 + x 2x2 - 2x4,

0 2 x2 2x dx = 2 (-1)2 = - 4

I = 48 ( 324 - 4 ) = 2 3 - 12 = (2 2 - 12)

so = 2, = -12 and ( + )2 = (-10)2 = 100.

Answer: (2)

Keep practising — Integrals

Ques. 2JEE 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. 3JEE 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. 4JEE 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. 5JEE 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. 6JEE 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. 7JEE 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
Ques. 8JEE Main 2025, 23 January (Morning Shift)

Let I(x)= dx(x-11) 1113(x+15) 1513. If I(37)-I(24)= 14 ( 1b 113- 1c 113 ), b,c N, then 3(b+c) is equal to

  1. 39
  2. 22
  3. 40
  4. 26
Ques. 9JEE Main 2025, 23 January (Evening Shift)

If I= 0 2 32x 32x+ 32x dx, then 02I x x x 4x+ 4x dx equals :

  1. 212
  2. 24
  3. 216
  4. 28
Ques. 10JEE Main 2025, 23 January (Evening Shift)

Let x3 x dx=g(x)+C, where C is the constant of integration. If 8 (g ( 2 )+g' ( 2 ) )= 3+ 2+ , , , Z, then + - equals :

  1. 47
  2. 55
  3. 62
  4. 48
Ques. 11JEE Main 2025, 22 January (Morning Shift)

Let for f(x)=7 8x+7 6x-3 4x-3 2x, I1= 0 /4f(x) dx and I2= 0 /4x f(x) dx. Then 7I1+12I2 is equal to :

  1. 2
  2. 1
  4. 2

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