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Integrals · Question 7

JEE Main 2026 (Online) 5th April Evening ShiftMCQMedium📈 Topic in 13% papers+4−1
1

Let (21-a+21+a ), f(a), (3a+3-a ) be in A.P. and be the minimum value of f(a). Then the value of the integral e( -1) e( ) dx (e2x-e-2x ) is :

  1. 12 e ( 43 )
  2. 14 e ( 43 )
  3. 12 e ( 85 )
  4. 14 e ( 85 )

Answer

(2) 14 e ( 43 )

Solution

A.P. 2f(a)= (21-a+21+a )+ (3a+3-a )=2 (2a+2-a )+ (3a+3-a )

By A.M. G.M.: 2a+2-a 2 and 3a+3-a 2 (equality at a=0)

2f(a) 6 f(a) 3 =3

I= e2 e3 dxe2x-e-2x= e2 e3 e2x dxe4x-1

Put e2x=u 2e2xdx=du; limits u=4 to u=9

I= 12 49 duu2-1= 14 [ e u-1u+1 ]49

I= 14 [ e 810- e 35 ]= 14 e ( 45 53 )

I= 14 e ( 43 )

Answer: (2)

JEE Main 2026 (Online) 5th April Evening Shift

Keep practising — Integrals

2

Let f : [1, ) R be a differentiable function defined as f(x)= 1x f(t) dt +(1-x) ( e x-1 )+e. Then the value of f(f(1)) is :

  1. (1+ee )
  2. (1+e)
  3. (1+e+ee )
  4. 1+2e
JEE Main 2026 (Online) 5th April Evening Shift
3

Let -22 (| x|+[x x] )dx=2(3- 2)+ , where [ ] is the greatest integer function. Then ( 2 ) equals:

  1. 1
  2. 2
  3. 4
  4. 8
JEE Main 2026 (Online) 4th April Morning Shift
4

If =1 and =1+i 2, where i= -1 are two roots of the equation x3+ax2+bx+c=0, a, b, c R, then -11 (x3+ax2+bx+c )dx is equal to :

  1. -2
  2. -4
  3. -8
  4. -10
JEE Main 2026 (Online) 4th April Evening Shift
5

The integral 01 -1 (1+x+x2 )dx is equal to :

  1. 2 -12+ 12 e ( 54 )+ 2
  2. 2 -12+ 12 e ( 54 )- 2
  3. 2 -12- 12 e ( 54 )+ 2
  4. 2 -12- 12 e ( 54 )- 2
JEE Main 2026 (Online) 4th April Evening Shift
6

Let [.] denote the greatest integer function. Then the value of 03 ( ex+e-x[x]! )dx is :

  1. e2+e3- 1e2- 1e3
  2. 12 (e2+e3- 1e2- 1e3 )
  3. e2+e3- 12e2- 12e3
  4. 12 (e2+e3 )- 1e2- 1e3
JEE Main 2026 (Online) 2nd April Morning Shift
7

If = 02 3 2 (x2+4 )dx+ 24 2x-4 dx, then 2 is equal to .

JEE Main 2026 (Online) 2nd April Morning Shift
8

The value of 020 ( 4 x+ 4 x )dx is equal to:

  1. 15 2
  2. 25
  3. 15
  4. 25 2
JEE Main 2026 (Online) 2nd April Evening Shift
9

Let f(x)= ( 16x+24x2+2x-15 )dx. If f(4)=14 e(3) and f(7)= e (2 3 ), , N, then + is equal to :

  1. 31
  2. 37
  3. 39
  4. 41
JEE Main 2026 (Online) 2nd April Evening Shift
10

Let f be a polynomial function such that f(x2 + 1) = x4 + 5x2 + 2, for all x R. Then 03 f(x) dx is equal to

  1. 332
  2. 53
  3. 272
  4. 413
JEE Main 2026 (Online) 28th January Morning Shift
11

If ( 1 - 5 2 x 5 x 2 x )dx = f(x) + C, where C is the constant of integration, then f ( 6 ) - f ( 4 ) is equal to

  1. 1 3(26 - 3)
  2. 4 3(8 - 6)
  3. 1 3(26 + 3)
  4. 2 3(4 + 6)
JEE Main 2026 (Online) 28th January Morning Shift

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