Integrals

JEE Main 2025 (Online) 4th April Evening ShiftMCQMedium📈 Topic in 5% papers+4−1

Ques. 1JEE Main 2025, 4 April (Evening Shift)

Let f(x)+2f ( 1x )=x2+5 and 2g(x)-3g ( 12 )=x, x>0. If = 12 f(x) dx, and = 12 g(x) dx, then the value of 9 + is :

Answer

(4) 11

Solution

f(x)+2f ( 1x )=x2+5 ... (i)

Replace x by 1x: f ( 1x )+2f(x)= 1x2+5 ... (ii)

2 (ii) - (i): 3f(x)= 2x2-x2+5

f(x)= 13 ( 2x2-x2+5 )

= 13 [- 2x- x33+5x ]12= 13 [ (-1- 83+10 )- (-2- 13+5 ) ]

= 13 ( 193- 83 )= 119 9 =11

For g, put x= 12: 2g ( 12 )-3g ( 12 )= 12 g ( 12 )=- 12

2g(x)+ 32=x g(x)= x2- 34

= [ x24- 3x4 ]12= (1- 32 )- ( 14- 34 )=0

9 + =11+0=11

Answer: (4)

Keep practising — Integrals

Ques. 2JEE Main 2025, 4 April (Evening Shift)

If ( 1+x2+x )10 ( 1+x2-x )9 dx= 1m ( ( 1+x2+x )n (n 1+x2-x ) )+C where C is the constant of integration and m, n N, then m+n is equal to .

Ques. 3JEE Main 2025, 3 April (Morning Shift)

Let f(x) = x3 3 - x2 dx. If 5f( 2) = -4, then f(1) is equal to

- 6 25
- 8 25
- 2 25
- 4 25

Ques. 4JEE Main 2025, 3 April (Morning Shift)

Let the domain of the function f(x) = 2 4 6 (3 + 4x - x2) be (a, b). If 0b-a [x2] dx = p - q - r, p, q, r N, (p, q, r) = 1, where [ ] is the greatest integer function, then p + q + r is equal to

Ques. 5JEE Main 2025, 3 April (Evening Shift)

The integral 0 8x dx4 2 x + 2 x is equal to

2 2
4 2
2
3 22

Ques. 6JEE Main 2025, 2 April (Morning Shift)

Let [.] denote the greatest integer function. If 0e3 [ 1ex-1 ] dx = - e 2, then 3 is equal to .

Ques. 7JEE Main 2025, 2 April (Evening Shift)

Let (a, b) be the point of intersection of the curve x2 = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral I = ab 9x21 + 5x dx is equal to :

Ques. 8JEE Main 2025, 2 April (Evening Shift)

4 01 ( 1 3 + x2 + 1 + x2 )dx - 3 e( 3) is equal to :