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Integrals

JEE Main 2024 (Online) 6th April Evening ShiftNumericalMedium📈 Topic in 67% papers+4
Ques. 1JEE Main 2024, 6 April (Evening Shift)

Let [t] denote the largest integer less than or equal to t. If 03 ( [x2 ] + [ x22 ] )dx = a + b 2 - 3 - 5 + c 6 - 7, where a, b, c Z, then a + b + c is equal to .

Answer

23

Solution

[x2 ] rises by 1 at each x = k (k = 1, 2, , 8); each rise adds a strip of area 1 (3 - k )

03 [x2 ]dx = k=18 (3 - k )

= 24 - (1 + 2 + 3 + 2 + 5 + 6 + 7 + 2 2 )

= 21 - 3 2 - 3 - 5 - 6 - 7

[ x22 ] rises by 1 at x = 2, 2, 6, 8:

03 [ x22 ]dx = 12 - ( 2 + 2 + 6 + 2 2 ) = 10 - 3 2 - 6

Sum = 31 - 6 2 - 3 - 5 - 2 6 - 7

a = 31, b = -6, c = -2 a + b + c = 23

Answer: 23

Keep practising — Integrals

Ques. 2JEE Main 2024, 5 April (Morning Shift)

The integral 0 /4 136 x3 x + 5 x dx is equal to :

  1. 3 - 50 e 2 + 20 e 5
  2. 3 - 25 e 2 + 10 e 5
  3. 3 - 10 e(2 2) + 10 e 5
  4. 3 - 30 e 2 + 20 e 5
Ques. 3JEE Main 2024, 5 April (Morning Shift)

The value of - 2y(1 + y)1 + 2 y dy is :

  1. 2
  2. 2
  3. 22
  4. 2 2
Ques. 4JEE Main 2024, 5 April (Evening Shift)

Let (m, n) = 01 xm - 1(1 - x)n - 1 dx, m, n > 0. If 01 (1 - x10 )20 dx = a (b, c), then 100(a + b + c) equals .

  1. 2012
  2. 1021
  3. 1120
  4. 2120
Ques. 5JEE Main 2024, 5 April (Evening Shift)

If f(t) = 0 2x dx1 - 2 t 2 x, 0 < t < , then the value of 0 2 2 dtf(t) equals .

Ques. 6JEE Main 2024, 4 April (Morning Shift)

Let f(x)= cases-2, & -2 x 0 \ x-2, & 0<x 2 cases and h(x)=f(|x|)+|f(x)|. Then -22 h(x) dx is equal to:

  1. 2
  2. 6
  3. 4
  4. 1
Ques. 7JEE Main 2024, 4 April (Morning Shift)

If 0 4 2x1+ x x dx= 1a e ( a3 )+ b 3, where a, b , then a+b is equal to .

Ques. 8JEE Main 2024, 4 April (Evening Shift)

If the value of the integral -11 x1 + 3x dx is 2 , then a value of is

  1. 2
  2. 4
  3. 3
  4. 6
Ques. 9JEE Main 2024, 4 April (Evening Shift)

If 5 x dx = x cosec x (cosec2 x + 32 ) + e | x2 | + C where , R and C is the constant of integration, then the value of 8( + ) equals .

Ques. 10JEE Main 2024, 1 February (Morning Shift)

The value of the integral 0 /4 x dx 4(2x)+ 4(2x) equals :

  1. 2 28
  2. 2 216
  3. 2 232
  4. 2 264
Ques. 11JEE Main 2024, 1 February (Morning Shift)

If - /2 /2 8 2 x dx (1+e x ) (1+ 4x )= + e(3+2 2), where , are integers, then 2+ 2 equals :

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