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Integrals

JEE Main 2024 (Online) 30th January Evening ShiftMCQMedium📈 Topic in 24% papers+4−1
Ques. 1JEE Main 2024, 30 January (Evening Shift)

Let y=f(x) be a thrice differentiable function in (-5,5). Let the tangents to the curve y=f(x) at (1,f(1)) and (3,f(3)) make angles /6 and /4, respectively with positive x-axis. If 27 13 ( (f'(t) )2+1 )f''(t) dt= + 3 where , are integers, then the value of + equals

  1. 26
  2. -16
  3. 36
  4. -14

Answer

(1) 26

Solution

Slope of tangent =f'(x): f'(1)= 6= 1 3, f'(3)= 4=1

Put f'(t)=z f''(t) dt=dz

13 ((f'(t))2+1 )f''(t) dt= 1/ 31(z2+1) dz= [ z33+z ]1/ 31

= 43- ( 19 3+ 1 3 )= 43- 10 327

27 ( 43- 10 327 )=36-10 3 =36, =-10

∴ + =26

Answer: (1)

Keep practising — Integrals

Ques. 2JEE Main 2024, 30 January (Evening Shift)

Let f: R R be defined as f(x)=ae2x+bex+cx. If f(0)=-1, f' ( e2 )=21 and 0 e4(f(x)-cx) dx= 392, then the value of |a+b+c| equals

  1. 16
  2. 12
  3. 8
  4. 10
Ques. 3JEE Main 2024, 29 January (Morning Shift)

For x (- 2, 2 ), if y(x)= cosec x+ xcosec x x+ x 2x dx, and x ( 2 )-y(x)=0 then y ( 4 ) is equal to

  1. - 1 2 -1 ( 1 2 )
  2. -1 ( 1 2 )
  3. 12 -1 ( 1 2 )
  4. 1 2 -1 (- 12 )
Ques. 4JEE Main 2024, 29 January (Morning Shift)

If the value of the integral - 2 2 ( x2 x1+ x+ 1+ 2x1+e x2023 )dx= 4( +a)-2, then the value of a is

  1. - 32
  2. 3
  3. 32
  4. 2
Ques. 5JEE Main 2024, 29 January (Evening Shift)

If 32x+ 32x 3x 3x (x- ) dx =A x- +B - x+C, where C is the integration constant, then AB is equal to

  1. 2
  2. 8 cosec(2 )
  3. 4 cosec(2 )
  4. 4
Ques. 6JEE Main 2024, 29 January (Evening Shift)

If /6 /3 1- 2x dx= + 2+ 3, where , and are rational numbers, then 3 +4 - is equal to .

Ques. 7JEE Main 2024, 27 January (Morning Shift)

If 01 1 3+x+ 1+x dx=a+b 2+c 3, where a, b, c are rational numbers, then 2a+3b-4c is equal to :

  1. 10
  2. 7
  3. 4
  4. 8
Ques. 8JEE Main 2024, 27 January (Morning Shift)

If (a, b) be the orthocentre of the triangle whose vertices are (1,2),(2,3) and (3,1), and I1= ab x (4x-x2 )dx, I2= ab (4x-x2 )dx, then 36 I1I2 is equal to :

  1. 80
  2. 72
  3. 66
  4. 88
Ques. 9JEE Main 2024, 27 January (Evening Shift)

The integral (x8-x2 )dx (x12+3x6+1 ) -1 (x3+ 1x3 ) is equal to :

  1. e ( | -1 (x3+ 1x3 ) | )1/3+C
  2. e ( | -1 (x3+ 1x3 ) | )+C
  3. e ( | -1 (x3+ 1x3 ) | )1/2+C
  4. e ( | -1 (x3+ 1x3 ) | )3+C
Ques. 10JEE Main 2024, 27 January (Evening Shift)

For 0<a<1, the value of the integral 0 dx1-2a x+a2 is :

  1. 2 +a2
  2. 2 -a2
  3. 1-a2
  4. 1+a2
Ques. 11JEE Main 2024, 27 January (Evening Shift)

Let f(x)= 0x g(t) e ( 1-t1+t )dt, where g is a continuous odd function. If - /2 /2 (f(x)+ x2 x1+ex )dx= ( )2- , then is equal to .

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