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Integrals

JEE Main 2024 (Online) 4th April Morning ShiftNumericalMedium📈 Topic in 24% papers+4
Ques. 1JEE 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 .

Answer

8

Solution

Since 2 2x = 1 - 2x and 2(1 + x x) = 2 + 2x, the integrand is 1 - 2x2 + 2x, so I = I1 - I2 with

I1 = 0 /4 dx2 + 2x and I2 = 0 /4 2x dx2 + 2x

In I2 the numerator is half the derivative of the denominator, hence I2 = 12 [ e(2 + 2x) ]0 /4 = 12 e 32

In I1 write 2x = 2 x1 + 2x and put x = t, so that 2x dx = dt:

I1 = 01 dt2t2 + 2t + 2 = 12 01 dt (t + 12 )2 + 34

= 1 3 [ -1 2t + 1 3 ]01 = 1 3 ( 3 - 6 ) = 6 3

Thus I = 12 e ( 23 ) + 6 3, which gives a = 2, b = 6.

a + b = 8

Answer: 8

Keep practising — Integrals

Ques. 2JEE 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. 3JEE 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. 4JEE 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. 5JEE 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 :

Ques. 6JEE Main 2024, 1 February (Evening Shift)

If 0 /3 4 x d x=a +b 3, where a and b are rational numbers, then 9 a+8 b is equal to :

  1. 2
  2. 1
  3. 3
  4. 32
Ques. 7JEE Main 2024, 1 February (Evening Shift)

The value of 01 (2x3-3x2-x+1 ) 13dx is equal to :

  1. -1
  2. 2
  3. 0
  4. 1
Ques. 8JEE Main 2024, 1 February (Evening Shift)

Let f:(0, ) and F(x)= 0x tf(t) dt. If F (x2 )=x4+x5, then r=112f (r2 ) is equal to .

Ques. 9JEE Main 2024, 31 January (Morning Shift)

If the integral 525 0 2 2x 112x (1+ 52x ) 12dx is equal to (n 2-64 ), then n is equal to .

Ques. 10JEE Main 2024, 31 January (Morning Shift)

Let f : R R be a function defined by f(x)= 4x4x+2 and M= f(a)f(1-a)x 4(x(1-x))dx, N= f(a)f(1-a) 4(x(1-x))dx; a 12. If M= N, , N, then the least value of 2+ 2 is equal to .

Ques. 11JEE Main 2024, 31 January (Evening Shift)

Let f, g:(0, ) R be two functions defined by f(x)= -xx (|t|-t2 )e-t2dt and g(x)= 0x2t1/2e-tdt. Then, the value of 9 (f ( e9 )+g ( e9 ) ) is equal to :

  1. 10
  2. 9
  3. 8
  4. 6

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