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Integrals · Question 23

JEE Main 2026 (Online) 23rd January Morning ShiftMCQMedium📈 Topic in 16% papers+4−1
1

Let f(x) = (2 - x2 ) ex ( 1 + x )(1 - x)3/2 dx. If f(0) = 0, then f ( 12 ) is equal to:

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

Answer

(3) 3e - 1

Solution

Try the form ex [g(x) + g'(x) ]dx = ex g(x) + C with g(x) = 1 + x1 - x.

g(x) = 1 - x2 1 + x (1 - x)3/2

g'(x) = 12 1 - x1 + x (1 - x) + (1 + x)(1 - x)2 = 1 1 + x (1 - x)3/2

g(x) + g'(x) = 2 - x2 1 + x (1 - x)3/2 (the given integrand without ex)

f(x) = ex 1 + x1 - x + C

f(0) = 0 1 + C = 0 C = -1

f ( 12 ) = e1/2 3/21/2 - 1 = 3e - 1

Answer: (3)

JEE Main 2026 (Online) 23rd January Morning Shift

Keep practising — Integrals

2

The value of the integral 24 5 24 dx1 + [3] 2x is :

  1. 3
  2. 18
  3. 6
  4. 12
JEE Main 2026 (Online) 23rd January Morning Shift
3

Let I(x) = 3 dx(4x + 6) ( 4x2 + 8x + 3 ) and I(0) = 34 + 20. If I ( 12 ) = a 2b + c, where a, b, c N, (a, b) = 1, then a + b + c is equal to :

  1. 30
  2. 29
  3. 28
  4. 31
JEE Main 2026 (Online) 23rd January Evening Shift
4

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JEE Main 2026 (Online) 23rd January Evening Shift
5

The value of - 2 2 ( 1[x] + 4 ) dx, where [ ] denotes the greatest integer function, is

  1. 160(21 - 1)
  2. 160( - 7)
  3. 760( - 3)
  4. 760(3 - 1)
JEE Main 2026 (Online) 22nd January Morning Shift
6

If ( x) -112( x) -52 dx = - p1q1( x) 92 - p2q2( x) 52 - p3q3( x) 12 + p4q4( x) -32 + C, where pi and qi are positive integers with (pi, qi) = 1 for i = 1, 2, 3, 4 and C is the constant of integration, then 15p1p2p3p4q1q2q3q4 is equal to

JEE Main 2026 (Online) 22nd January Morning Shift
7

Let [.] be the greatest integer function. If = 064 (x1/3 - [x1/3] )dx, then 1 0 ( 2 6 + 6 )d is equal to .

JEE Main 2026 (Online) 22nd January Evening Shift
8

The value of - /6 /6 ( + 4x111 - (|x| + /6) )dx is equal to:

  1. 8
  2. 4
  3. 2
  4. 6
JEE Main 2026 (Online) 21st January Morning Shift
9

6 0 |( 3x + 2x + x)| dx is equal to .

JEE Main 2026 (Online) 21st January Morning Shift
10

Let [ ] denote the greatest integer function and f(x) = n 1n3 k=1n [ k23x ]. Then 12 j=1 f(j) is equal to .

JEE Main 2026 (Online) 21st January Evening Shift
11

If 01 4 -1 (1 - 2x + 4x2 )dx = a -1(2) - b e(5), where a, b N, then (2a + b) is equal to .

JEE Main 2026 (Online) 21st January Evening Shift

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