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Integrals

JEE Main 2025 (Online) 28th January Evening ShiftMCQMedium📈 Topic in 12% papers+4−1
Ques. 1JEE 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

Answer

(3) 310

Solution

Since g(x) = 0x t f(t) dt, g'(x) = x f(x). Differentiating g(x3) = x6 + x7:

g'(x3) 3x2 = 6x5 + 7x6

x3 f(x3) 3x2 = 6x5 + 7x6

f(x3) = 6x5 + 7x63x5 = 2 + 7x3

Hence

r=115 f(r3) = r=115 (2 + 7r3 )

= 30 + 73 15 162

= 30 + 73 120 = 30 + 280 = 310

Answer: (3)

Keep practising — Integrals

Ques. 2JEE 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. 3JEE 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. 4JEE 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. 5JEE 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. 6JEE 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. 7JEE 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. 8JEE 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
Ques. 9JEE Main 2025, 22 January (Evening Shift)

If x ( x -1x 1-x2+ -1x(1-x2)3/2+ x1-x2 )dx=g(x)+C, where C is the constant of integration, then g ( 12 ) equals :

  1. 6 e3
  2. 6 e2
  3. 4 e3
  4. 4 e2
Ques. 10JEE Main 2024, 9 April (Morning Shift)

Let 2 - x3 + x dx = 12 ( x + e| x + x| ) + C, where C is the constant of integration. Then + is equal to :

  1. 3
  2. 7
  3. 1
  4. 4
Ques. 11JEE Main 2024, 9 April (Morning Shift)

Let n ( n n4 + 1 - 2n (n2 + 1 ) n4 + 1 . . + n n4 + 16 - 8n (n2 + 4 ) n4 + 16 + . . + n n4 + n4 - 2n n2 (n2 + n2 ) n4 + n4 ) be k, using only the principal values of the inverse trigonometric functions. Then k2 is equal to .

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