PYQSide ExamsMy practice

Integrals · Question 25

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

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

Answer

(4) 31

Solution

4x2 + 8x + 3 = (2x + 1)(2x + 3) and 4x + 6 = 2(2x + 3)

Put t = 2x + 3 dx = dt2, 2x + 1 = t - 2

I = 3 dt22t t(t - 2) = 34 dtt2 1 - 2t

Now put 1 - 2t = u2 2t2 dt = 2u du dtt2 = u du

I = 34 u duu = 34u + C

I(x) = 34 2x + 12x + 3 + C

I(0) = 34 1 3 + C = 34 + C

I(0) = 34 + 20 C = 20

I ( 12 ) = 34 24 + 20 = 34 22 + 20 = 3 28 + 20

a = 3, b = 8, c = 20 (and (3, 8) = 1)

a + b + c = 3 + 8 + 20 = 31

Answer: (4)

JEE Main 2026 (Online) 23rd January Evening Shift

Keep practising — Integrals

2

The number of elements in the set S = \x : x [0, 100] and 0x t2 (x - t) dt = x2 \ is

JEE Main 2026 (Online) 23rd January Evening Shift
3

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
4

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
5

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
6

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
7

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

JEE Main 2026 (Online) 21st January Morning Shift
8

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
9

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
10

The integral -1 32 ( | 2 x ( x) | )dx is equal to:

  1. 2 + 3
  2. 4 +
  3. 1 + 3
  4. 3 + 2
JEE Main 2025 (Online) 8th April Evening Shift
11

Let f(x) be a positive function and I1 = - 121 2x f(2x(1 - 2x)) dx and I2 = -12 f(x(1 - x)) dx. Then the value of I2I1 is equal to

  1. 12
  2. 9
  3. 6
  4. 4
JEE Main 2025 (Online) 8th April Evening Shift

All Integrals PYQs →

PDFDownload Integrals PYQs as PDF Free, with answer key and a QR code to every solution — print it or study offline⬇
All JEE Main Mathematics chapters
Class 11SetsRelations and FunctionsTrigonometric FunctionsComplex Numbers and Quadratic EquationsPermutations and CombinationsBinomial TheoremSequences and SeriesStraight LinesConic SectionsIntroduction to Three Dimensional GeometryLimits and DerivativesStatisticsProbability
Class 12Relations and Functions (Class 12)Inverse Trigonometric FunctionsMatricesDeterminantsContinuity and DifferentiabilityApplication of DerivativesIntegralsApplication of IntegralsDifferential EquationsVector AlgebraThree Dimensional GeometryProbability (Class 12)