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JEE Indefinite and Definite Integration PYQs with Solutions

182 previous year questions from Indefinite and Definite Integration (JEE Main & Advanced Mathematics) with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

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About Indefinite and Definite Integration PYQs

Indefinite and Definite Integration has 182 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 182 questions in this list come from Black Book Advanced Problems in Mathematics (182). Each question opens on its own page with the options, the correct answer and the worked solution where available.

Indefinite and Definite Integration questions with solutions

1

**Paragraph for Question Nos. 1 to 2** Let f(x) = ∫ x^2 ^2 x (2x + 6 x - 2x ^2 x) dx and f(x) passes through the point (π, 0) If f : R - (2n + 1)(π)/(2) arrow R then f(x) be a :

Black Book Advanced Problems in Mathematics Solution View → 2

**Paragraph for Question Nos. 1 to 2** Let f(x) = ∫ x^2 ^2 x (2x + 6 x - 2x ^2 x) dx and f(x) passes through the point (π, 0) The number of solution(s) of the equation f(x) = x^3

Black Book Advanced Problems in Mathematics Solution View → 3

**Paragraph for Question Nos. 3 to 4** Let f(x) be a twice differentiable function defined on (-∞, ∞) such that f(x) = f(2 - x) and f'((1)/(2)) = f'((1)/(4)) = 0 . Then The

Black Book Advanced Problems in Mathematics Solution View → 4

**Paragraph for Question Nos. 3 to 4** Let f(x) be a twice differentiable function defined on (-∞, ∞) such that f(x) = f(2 - x) and f'((1)/(2)) = f'((1)/(4)) = 0 . Then ∫_ -1 ^1

Black Book Advanced Problems in Mathematics Solution View → 5

**Paragraph for Question Nos. 3 to 4** Let f(x) be a twice differentiable function defined on (-∞, ∞) such that f(x) = f(2 - x) and f'((1)/(2)) = f'((1)/(4)) = 0 . Then ∫_0^2 f(1

Black Book Advanced Problems in Mathematics Solution View → 6

**Paragraph for Question Nos. 6 to 8** Consider the function f(x) and g(x) , both defined from R arrow Rf(x) = (x^3)/(2) + 1 - x ∫_0^x g(t) dt and g(x) = x - ∫_0^1 f(t) dt , then

Black Book Advanced Problems in Mathematics Solution View → 7

**Paragraph for Question Nos. 6 to 8** Consider the function f(x) and g(x) , both defined from R arrow Rf(x) = (x^3)/(2) + 1 - x ∫_0^x g(t) dt and g(x) = x - ∫_0^1 f(t) dt , then

Black Book Advanced Problems in Mathematics Solution View → 8

**Paragraph for Question Nos. 6 to 8** Consider the function f(x) and g(x) , both defined from R arrow Rf(x) = (x^3)/(2) + 1 - x ∫_0^x g(t) dt and g(x) = x - ∫_0^1 f(t) dt , then

Black Book Advanced Problems in Mathematics Solution View → 9

**Paragraph for Question Nos. 9 to 11** Let f(x) be function defined on such that f(1) = 0 and for any a ∈ (0, 1] , ∫_0^a f(x) dx - ∫_a^1 f(x) dx = 2f(a) + 3a + b where b is

Black Book Advanced Problems in Mathematics Solution View → 10

**Paragraph for Question Nos. 9 to 11** Let f(x) be function defined on such that f(1) = 0 and for any a ∈ (0, 1] , ∫_0^a f(x) dx - ∫_a^1 f(x) dx = 2f(a) + 3a + b where b is

Black Book Advanced Problems in Mathematics Solution View → 11

**Paragraph for Question Nos. 9 to 11** Let f(x) be function defined on such that f(1) = 0 and for any a ∈ (0, 1] , ∫_0^a f(x) dx - ∫_a^1 f(x) dx = 2f(a) + 3a + b where b is

Black Book Advanced Problems in Mathematics Solution View → 12

**Paragraph for Question Nos. 12 to 13** Let f_0(x) = x and for n ≥ 0 and x > 0 Let f_ n+1 (x) = ∫_0^x f_n(t) dt then : f_3(x) equals :

Black Book Advanced Problems in Mathematics Solution View → 13

**Paragraph for Question Nos. 12 to 13** Let f_0(x) = x and for n ≥ 0 and x > 0 Let f_ n+1 (x) = ∫_0^x f_n(t) dt then : Value of _ n → ∞ ((n!) f_n(1))/( (n))

Black Book Advanced Problems in Mathematics Solution View → 14

**Paragraph for Question Nos. 14 to 15** Let f : R arrow [-(3)/(4), ∞) be a surjective quadratic function with line of symmetry 2x - 1 = 0 and f(1) = 1 If g(x) = (f(x) +

Black Book Advanced Problems in Mathematics Solution View → 15

**Paragraph for Question Nos. 14 to 15** Let f : R arrow [-(3)/(4), ∞) be a surjective quadratic function with line of symmetry 2x - 1 = 0 and f(1) = 1 ∫ (e^x)/(f(e^x)) dx

Black Book Advanced Problems in Mathematics Solution View → 16

**Paragraph for Question Nos. 16 to 17** Let g(x) = x^c e^ cx and f(x) = ∫_0^x t e^ 2t (1 + 3t^2)^ 1/2 dt . If L = _ x → ∞ (f'(x))/(g'(x)) is non-zero finite number then : The

Black Book Advanced Problems in Mathematics Solution View → 17

**Paragraph for Question Nos. 16 to 17** Let g(x) = x^c e^ cx and f(x) = ∫_0^x t e^ 2t (1 + 3t^2)^ 1/2 dt . If L = _ x → ∞ (f'(x))/(g'(x)) is non-zero finite number then : The

Black Book Advanced Problems in Mathematics Solution View → 18

Match the following columns: Column-I Column-II (A) _ n → ∞ 4 [ (1)/(n^2) e^ (1)/(n) + (2)/(n^2) e^ (2)/(n) + (3)/(n^2) e^ (3)/(n) + … + (1)/(n) e^ (n)/(n) ] = (P) 0 (B) ∫_0^1 (

Black Book Advanced Problems in Mathematics Solution View → 19

Match the following ∫ f(x) dx is equal to, if Column-I Column-II (A) f(x) = 1 (x^2 + 1) x^2 + 2 (P) x^5 5(1 - x^4)^ 5/2 + C (B) f(x) = 1 (x + 2) x^2 + 6x + 7 (Q) ^ -1 ( x + 1 (x +

Black Book Advanced Problems in Mathematics Solution View → 20

Match the following columns: Column-I Column-II (A) ∫_0^ π /2 ( x)/((1 + x)(2 + x)) dx = (P) (π)/(6) (B) ∫_0^ 41 π /4 | x| dx = (Q) 20 + 1 2 (C) ∫_ -1/2 ^ 1/2 [ [ x] + ( (1 +

Black Book Advanced Problems in Mathematics Solution View → 21

Match the following columns: Column-I Column-II (A) If quadratic equation 3x^2 + ax + 1 = 0 and 2x^2 + bx + 1 = 0 have a common root then value of 5ab - 2a^2 - 3b^2 = (P) 6 (B)

Black Book Advanced Problems in Mathematics Solution View → 22

Match the following columns: Column-I Column-II (A) ∫_0^ 1.5 [x^2] dx (P) - π (B) ∫_0^4 \ x \ dx (where \ x\ denotes the fractional part of x ) (Q) 4( 2 - 1) (C) ∫_0^ 2 π [ x + x]

Black Book Advanced Problems in Mathematics Solution View → 23

∫ x + ( 3x)^2 1 - 9x^2 dx = (1)/(k_1) ( 1 - 9x^2 + ( ^ -1 3x)^ k_2 ) + C , then k_1^2 + k_2^2 = *(where C is an arbitrary constant.)*

Black Book Advanced Problems in Mathematics Solution View → 24

If ∫_0^∞ (x^3)/((a^2 + x^2)^5) dx = (1)/(k a^6) , then find the value of (k)/(8) .

Black Book Advanced Problems in Mathematics Solution View → 25

Let f(x) = x x ; x ∈ [ (3π)/(2), 2 π ] and g(x) be its inverse. If ∫_0^ 2 π g(x) dx = α π^2 + β π + γ, where α, β and γ ∈ R , then find the value of 2(α + β + γ).

Black Book Advanced Problems in Mathematics Solution View → 26

If ∫ (x^6 + x^4 + x^2) 2x^4 + 3x^2 + 6 dx = (α x^6 + β x^4 + γ x^2)^ 3/2 18 + C where C is constant, then find the value of (β + γ - α) .

Black Book Advanced Problems in Mathematics Solution View → 27

If the value of the definite integral ∫_ -1 ^1 ^ -1 ( 1 1 - x^2 ) ( ^ -1 x 1 - x^2 ) |x| dx = π^2( a - b ) c where a, b, c ∈ N in their lowest form, then find the value of (a + b

Black Book Advanced Problems in Mathematics Solution View → 28

The value of ∫ ( x)/( ^2 x + x + 1) dx = x - 2 A ^ -1 ( 2 x + 1 A ) + C Then the value of A is:

Black Book Advanced Problems in Mathematics Solution View → 29

Let ∫_0^1 4x^3(1 + (x^4)^ 2010 ) (1 + x^4)^ 2012 dx = (λ)/(μ) where λ and μ are relatively prime positive integers. Find unit digit of μ .

Black Book Advanced Problems in Mathematics Solution View → 30

Let ∫_1^ 3 ( x^ 2x^2 + 1 + (x^ x^ 2x^2 + 1 ) ) dx = N . Find the value of (N - 6) .

Black Book Advanced Problems in Mathematics Solution View → 31

If ∫ (dx)/( ^3 x - ^3 x) = A ^ -1 (f(x)) + B | 2 + f(x) 2 - f(x) | + C where f(x) = x + x find the value of (12A + 9 2 B) - 3 .

Black Book Advanced Problems in Mathematics Solution View → 32

Find the value of |a| for which the area of triangle included between the coordinate axes and any tangent to the curve x^a y = λ^a is constant (where λ is constant.)

Black Book Advanced Problems in Mathematics Solution View → 33

Let I = ∫_0^π x^6 (π - x)^8 dx , then π^ 15 15 9 I =

Black Book Advanced Problems in Mathematics Solution View → 34

If maximum value of ∫_0^1 (f(x))^3 dx under the condition -1 ≤ f(x) ≤ 1 ; ∫_0^1 f(x) dx = 0 is (p)/(q) (where p and q are relatively prime positive integers.). Find p + q .

Black Book Advanced Problems in Mathematics Solution View → 35

Let a differentiable function f(x) satisfies f(x) · f'(-x) = f(-x) · f'(x) and f(0) = 1 . Find the value of ∫_ -2 ^2 (dx)/(1 + f(x)) .

Black Book Advanced Problems in Mathematics Solution View → 36

If \ x\ denotes the fractional part of x , then I = ∫_0^ 100 \ x \ dx , then the value of (9I)/(155) is:

Black Book Advanced Problems in Mathematics Solution View → 37

Let I_n = ∫_0^π ( ((n + (1)/(2))x))/( ((x)/(2))) dx where n ∈ W . If I_1^2 + I_2^2 + I_3^2 + … + I_ 20 ^2 = mπ^2 , then find the largest prime factor of m .

Black Book Advanced Problems in Mathematics Solution View → 38

If M be the maximum value of 72 ∫_0^y x^4 + (y - y^2)^2 dx for y ∈ , then find (M)/(6) .

Black Book Advanced Problems in Mathematics Solution View → 39

Find the number of points where f(θ) = ∫_ -1 ^1 ( θ dx)/(1 - 2x θ + x^2) is discontinuous where θ ∈ [0, 2π] .

Black Book Advanced Problems in Mathematics Solution View → 40

Find the value of _ n → ∞ 1 n ( 1 + 1 2 + 1 3 + … + 1 n ).

Black Book Advanced Problems in Mathematics Solution View → 41

The maximum value of ∫_ -π/2 ^ 3π/2 x · f(x) dx , subject to the condition |f(x)| ≤ 5 is M , then (M)/(10) is equal to:

Black Book Advanced Problems in Mathematics Solution View → 42

Given a function g , continuous everywhere such that g(1) = 5 and ∫_0^1 g(t) dt = 2 . If f(x) = (1)/(2) ∫_0^x (x - t)^2 g(t) dt , then find the value of f'''(1) + f''(1) .

Black Book Advanced Problems in Mathematics Solution View → 43

If f(n) = (1)/(π) ∫_0^ π/2 ( ^2(nθ) dθ)/( ^2 θ), n ∈ N , then evaluate (f(15) + f(3))/(f(12) - f(10)) .

Black Book Advanced Problems in Mathematics Solution View → 44

Let f(2 - x) = f(2 + x) and f(4 - x) = f(4 + x) . Function f(x) satisfies ∫_0^2 f(x) dx = 5 . If ∫_0^ 50 f(x) dx = I . Find [ I - 3] . *(where [·] denotes greatest integer

Black Book Advanced Problems in Mathematics Solution View → 45

Let I_n = ∫_ -1 ^1 |x| ( 1 + x + (x^2)/(2) + (x^3)/(3) + … + x^ 2n 2n ) dx . If _ n → ∞ I_n can be expressed as rational (p)/(q) in its lowest form, then find the value of

Black Book Advanced Problems in Mathematics Solution View → 46

Let _ n → ∞ n^ -(1)/(2)(1 + (1)/(n)) · (1^1 · 2^2 · 3^3 … n^n)^ (1)/(n^2) = e^ (-p)/(q) where p and q are relative prime positive integers. Find the value of |p + q| .

Black Book Advanced Problems in Mathematics Solution View → 47

If ∫_a^b | x| dx = 8 and ∫_0^ a+b | x| dx = 9 then the value of 1 2 π ∫_a^b x x dx is:

Black Book Advanced Problems in Mathematics Solution View → 48

If f(x), g(x), h(x) and φ(x) are polynomial in x , ( ∫_1^x f(x) h(x) dx ) ( ∫_1^x g(x) φ(x) dx ) - ( ∫_1^x f(x) φ(x) dx ) ( ∫_1^x g(x) h(x) dx ) is divisible by (x - 1)^λ . Find

Black Book Advanced Problems in Mathematics Solution View → 49

If ∫_0^1 (3x^2 - 3x + 1) (x^3 - 3x^2 + 4x - 2) dx = a (b) , where a and b are positive integers. Find the value of (a + b) .

Black Book Advanced Problems in Mathematics Solution View → 50

Let f(x) = ∫_0^x e^ x-y f'(y) dy - (x^2 - x + 1)e^x . Find the number of roots of the equation f(x) = 0 .

Black Book Advanced Problems in Mathematics Solution View → 51

For a positive integer n , let I_n = ∫_ -π ^π ( (π)/(2) - |x| ) nx dx . Find the value of [I_1 + I_2 + I_3 + I_4] where [·] denotes greatest integer function. ***

Black Book Advanced Problems in Mathematics Solution View → 52

∫ dx (1 + x )^8 = - 1 3(1 + x )^ k_1 + 2 7(1 + x )^ k_2 + C , then :

Black Book Advanced Problems in Mathematics Solution View → 53

If ∫_ -α ^ α ( e^x + x (x + 1 + x^2 ) ) dx > (3)/(2) , then possible value of α can be :

Black Book Advanced Problems in Mathematics Solution View → 54

For a > 0 , if I = ∫ (x)/(a^3 - x^3) dx = A ^ -1 ( x^ 3/2 B ) + C , where C is any arbitrary constant, then :

Black Book Advanced Problems in Mathematics Solution View → 55

Let ∫ x x · ^3 x dx = (1)/(2)(x · f(x) - g(x)) + k , then :

Black Book Advanced Problems in Mathematics Solution View → 56

If ∫ ( 3θ + θ) θ e^ θ dθ = (A ^3 θ + B ^2 θ + C θ + D θ + E) e^ θ + F , then :

Black Book Advanced Problems in Mathematics Solution View → 57

For a > 0 , if I = ∫ (x)/(a^3 - x^3) dx = A ^ -1 ( x^ 3/2 B ) + C , where C is any arbitrary constant, then :

Black Book Advanced Problems in Mathematics Solution View → 58

If f(θ) = _ n → ∞ ∑_ r=0 ^ nθ 2r n (3θ n - 2r)(nθ + 2r) then :

Black Book Advanced Problems in Mathematics Solution View → 59

If f(x+y) = f(x)f(y) for all x, y and f(0) ≠ 0 , and F(x) = (f(x))/(1 + (f(x))^2) then :

Black Book Advanced Problems in Mathematics Solution View → 60

Let J = ∫_ -1 ^2 ( ^ -1 (1)/(x) + ^ -1 x ) dx, K = ∫_ -2π ^ 7π ( x)/(| x|) dx . Then which of the following alternative(s) is/are correct ?

Black Book Advanced Problems in Mathematics Solution View → 61

Which of the following function(s) is/are even ?

Black Book Advanced Problems in Mathematics Solution View → 62

Let l_1 = _ x → ∞ (x - ^2 x)/(x + x) and l_2 = _ h → 0^+ ∫_ -1 ^1 (h dx)/(h^2 + x^2) . Then :

Black Book Advanced Problems in Mathematics Solution View → 63

For a > 0 , if I = ∫ (x)/(a^3 - x^3) dx = A ^ -1 ( x^ 3/2 B ) + C , where C is any arbitrary constant, then :

Black Book Advanced Problems in Mathematics Solution View → 64

If ∫ (dx)/(1 - ^4 x) = a x + b ^ -1 (c x) + D , then :

Black Book Advanced Problems in Mathematics Solution View → 65

The value of definite integral : ∫_ -2014 ^ 2014 dx 1 + ^ 2015 (x) + 1 + ^ 4030 (x) equals :

Black Book Advanced Problems in Mathematics Solution View → 66

Let L = _ n → ∞ ∫_a^∞ (n dx)/(1 + n^2 x^2) where a ∈ R then L can be :

Black Book Advanced Problems in Mathematics Solution View → 67

Let I = ∫_0^1 1 + x 1 - x dx and J = ∫_0^1 1 - x 1 + x dx then correct statement(s) is/are :

Black Book Advanced Problems in Mathematics Solution View → 68

∫ a^x ( x + a · ( (x)/(e) )^x ) dx =

Black Book Advanced Problems in Mathematics Solution View → 69

The value of : _ n → ∞ ( 1 n n+1 + 1 n n+2 + 1 n n+3 + … + 1 n 2n ) is :

Black Book Advanced Problems in Mathematics Solution View → 70

If ∫ ( x)/( (x - α)) dx = Ax + B (x - α) + C , then value of (A, B) is :

Black Book Advanced Problems in Mathematics Solution View → 71

The value of the integral ∫_0^2 ( (x^2 + 2))/((x + 2)^2) dx is :

Black Book Advanced Problems in Mathematics Solution View → 72

If I_1 = ∫_0^1 (1 + x^8)/(1 + x^4) dx and I_2 = ∫_0^1 (1 + x^9)/(1 + x^3) dx , then :

Black Book Advanced Problems in Mathematics Solution View → 73

Let f:(0, 1) arrow (0, 1) be a differentiable function such that f'(x) ≠ 0 for all x ∈ (0, 1) and f((1)/(2)) = 3 2 . Suppose for all x , _ t → x ∫_0^t 1 - (f(s))^2 ds - ∫_0^x 1

Black Book Advanced Problems in Mathematics Solution View → 74

If f(θ) = (4)/(3) (1 - ^6 θ - ^6 θ) , then _ n → ∞ (1)/(n) [ f((1)/(n)) + f((2)/(n)) + f((3)/(n)) + … + f((n)/(n)) ] =

Black Book Advanced Problems in Mathematics Solution View → 75

The value of ∫_0^1 ((x^6 - x^3))/((2x^3 + 1)^3) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View → 76

2 ∫_0^ 1 2 ^ -1 x x dx - ∫_0^1 ^ -1 x x dx =

Black Book Advanced Problems in Mathematics Solution View → 77

Let f(x) be a differentiable function such that f(x) = x^2 + ∫_0^x e^ -t f(x - t) dt , then ∫_0^1 f(x) dx =

Black Book Advanced Problems in Mathematics Solution View → 78

If f'(x) = f(x) + ∫_0^1 f(x)dx and given f(0) = 1 , then ∫ f(x)dx is equal to: *(where C is an arbitrary constant.)*

Black Book Advanced Problems in Mathematics Solution View → 79

For any x ∈ R , and f be a continuous function. Let I_1 = ∫_ ^2 x ^ 1+ ^2 x t f(t(2-t))dt , I_2 = ∫_ ^2 x ^ 1+ ^2 x f(t(2-t))dt , then I_1 =

Black Book Advanced Problems in Mathematics Solution View → 80

If the integral ∫ (5 x)/( x - 2) dx = x + a | x - 2 x| + C , then ' a ' is equal to :

Black Book Advanced Problems in Mathematics Solution View → 81

∫ (2 + x ) dx (x + 1 + x )^2 is equal to : *(where C is an arbitrary constant.)*

Black Book Advanced Problems in Mathematics Solution View → 82

Evaluate ∫ x + 2 - x^2 1 - x 2 - x^2 1 - x^2 dx ; x ∈ (0, 1) :

Black Book Advanced Problems in Mathematics Solution View → 83

∫ dx 1 - ^2 x = (1)/(λ) ^ -1 (λ x) + C , then λ =

Black Book Advanced Problems in Mathematics Solution View → 84

∫ dx x^ 5/2 (x + 1)^ 7/2 is equal to :

Black Book Advanced Problems in Mathematics Solution View → 85

If I_n = ∫ ( x)^n dx; n ∈ N , then 5I_4 - 6I_6 is equal to:

Black Book Advanced Problems in Mathematics Solution View → 86

∫ (x^2)/((a + bx)^2) dx equals to :

Black Book Advanced Problems in Mathematics Solution View → 87

∫ 8x^ 43 + 13x^ 38 (x^ 13 + x^5 + 1)^4 dx =

Black Book Advanced Problems in Mathematics Solution View → 88

∫ ( ( 6x + 6 4x + 15 2x + 10)/(10 ^2 x + 5 x 3x + x 5x) ) dx = f(x) + C , then f(10) is equal to:

Black Book Advanced Problems in Mathematics Solution View → 89

∫ (1 + x - x^ -1 ) e^ x + x^ -1 dx =

Black Book Advanced Problems in Mathematics Solution View → 90

If ∫ e^x ( (2 x)/(1 + x) + cosec^2 ( x + (π)/(4) ) ) dx = e^x · g(x) + K , then g((5π)/(4)) =

Black Book Advanced Problems in Mathematics Solution View → 91

∫ e^ x x + x ( (x^4 ^3 x - x x + x)/(x^2 ^2 x) ) dx =

Black Book Advanced Problems in Mathematics Solution View → 92

The value of the definite integral ∫_0^1 1 + x + x+x^2 x + 1 + x dx is :

Black Book Advanced Problems in Mathematics Solution View → 93

∫ x^ x^2+1 (2 x + 1) dx

Black Book Advanced Problems in Mathematics Solution View → 94

If ∫ cosec^2 x - 2010 ^ 2010 x dx = - f(x) (g(x))^ 2010 + C ; where f((π)/(4)) = 1 ; then the number of solutions of the equation (f(x))/(g(x)) = \ x\ in [0, 2π] is/are : *(where

Black Book Advanced Problems in Mathematics Solution View → 95

∫ x^x ( ( x)^2 + x + (1)/(x) ) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View → 96

If I = ∫ x^2 - 1 x^3 2x^4 - 2x^2 + 1 dx is equal to:

Black Book Advanced Problems in Mathematics Solution View → 97

I = ∫ ( ( x - 1)/(( x)^2 + 1) )^2 dx is equal to:

Black Book Advanced Problems in Mathematics Solution View → 98

I = ∫ dx (x-1)^3(x+2)^5 = k (x-1)/(x+2) + C , then ' k ' is equal to :

Black Book Advanced Problems in Mathematics Solution View → 99

∫ (1-x^7)/(x(1+x^7)) dx = P |x| + Q |x^7 + 1| + C , then :

Black Book Advanced Problems in Mathematics Solution View → 100

I = ∫ ( ^8 x - ^8 x)/(1 - 2 ^2 x ^2 x) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View → 101

I = ∫ ( 2x)^ 1/3 d( ^ 1/3 x) ^ 2/3 x + ^ 2/3 x =

Black Book Advanced Problems in Mathematics Solution View → 102

∫ (2012)^ 2x 1 - (2012)^ 2x (2012)^ ^ -1 (2012)^x dx =

Black Book Advanced Problems in Mathematics Solution View → 103

∫ (x + 2) dx (x^2 + 3x + 3) x + 1 is equal to : *(where C is arbitrary constant.)*

Black Book Advanced Problems in Mathematics Solution View → 104

∫ ( (f(x)g'(x) - f'(x)g(x))/(f(x)g(x)) ) ( (g(x)) - (f(x))) dx is equal to:

Black Book Advanced Problems in Mathematics Solution View → 105

∫ [ e^x ( x + (2)/(x) - (1)/(x^2) ) ] dx =

Black Book Advanced Problems in Mathematics Solution View → 106

Maximum value of the function f(x) = π^2 ∫_0^1 t (x + π t) dt over all real number x :

Black Book Advanced Problems in Mathematics Solution View → 107

Let ' f ' is a function, continuous on such that f(x) ≤ 5 \; ∀ x ∈ and f(x) ≤ (2)/(x) \; ∀ x ∈ [(1)/(2), 1] then the smallest ' a ' for which ∫_0^1 f(x) dx ≤ a holds for all ' f '

Black Book Advanced Problems in Mathematics Solution View → 108

Let I_n = ∫_1^e ( x)^n d(x^2) , then the value of 2I_n + nI_ n-1 equals to :

Black Book Advanced Problems in Mathematics Solution View → 109

Let a function f: R arrow R be defined as f(x) = x + x . The value of ∫_0^ 2π f^ -1 (x) dx will be :

Black Book Advanced Problems in Mathematics Solution View → 110

The value of the definite integral ∫_ -1 ^1 e^ -x^4 (2 + (x + x^2+1 ) + 5x^3 - 8x^4) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View → 111

∫_ -10 ^0 (2[x])/(3x - [x]) dx is equal to (where [*] denotes greatest integer function.)

Black Book Advanced Problems in Mathematics Solution View → 112

If f(x) = (x)/(1 + ( x)( x)… ∞) \; ∀ x ∈ [1, ∞) then ∫_1^ 2e f(x) dx is:

Black Book Advanced Problems in Mathematics Solution View → 113

∫_0^2 ((y^2 - 4y + 5) (y - 2))/((2y^2 - 8y + 11)) dy is equal to :

Black Book Advanced Problems in Mathematics Solution View → 114

Let (d)/(dx) F(x) = ( e^ x x ), x > 0 . If ∫_1^4 (3)/(x) e^ x^3 dx = F(k) - F(1) , then one of the possible values of k , is:

Black Book Advanced Problems in Mathematics Solution View → 115

Value of _ h → 0 ∫_0^ π+h x^2 e^ -x^2 dx - ∫_0^π x^2 e^ -x^2 dx h is equal to :

Black Book Advanced Problems in Mathematics Solution View → 116

Let f: R^+ arrow R be a differentiable function with f(1) = 3 and satisfying : ∫_1^ xy f(t) dt = y ∫_1^x f(t) dt + x ∫_1^y f(t) dt \;∀ x, y ∈ R^+ , then f(e) =

Black Book Advanced Problems in Mathematics Solution View → 117

If [ · ] denotes the greatest integer function, then the integral ∫_0^ π/2 e^ x - [ x] d( ^2 x - [ ^2 x]) is λ , then [λ - 1] is equal to :

Black Book Advanced Problems in Mathematics Solution View → 118

Calculate the reciprocal of the limit _ x → ∞ ∫_0^x e^ t^2 - x^2 dt

Black Book Advanced Problems in Mathematics Solution View → 119

Let L = _ n → ∞ ( ((2 · 1+n))/(1^2+n · 1+n^2) + ((2 · 2+n))/(2^2+n · 2+n^2) + ((2 · 3+n))/(3^2+n · 3+n^2) + … + ((2 · n+n))/(3n^2) ) then value of e^L is :

Black Book Advanced Problems in Mathematics Solution View → 120

The value of the definite integral ∫_0^2 ( 1+x^3 + x^2+2x ) dx is :

Black Book Advanced Problems in Mathematics Solution View → 121

The value of the definite integral ∫_0^∞ ( x)/(x^2+4) dx is :

Black Book Advanced Problems in Mathematics Solution View → 122

The value of the definite integral ∫_0^ 10 ((x-5) + (x-5)^2 + (x-5)^3) dx is :

Black Book Advanced Problems in Mathematics Solution View → 123

The value of definite integral ∫_0^∞ (dx)/((1+x^9)(1+x^2)) equals to :

Black Book Advanced Problems in Mathematics Solution View → 124

The value of the definite integral ∫_0^ π/2 ( (1 + 3x)/(1 + 2 x) ) dx equals to :

Black Book Advanced Problems in Mathematics Solution View → 125

The value of _ x → ∞ ∫_0^x ( ^ -1 x)^2 dx x^2+1 =

Black Book Advanced Problems in Mathematics Solution View → 126

If ∫_0^1 ( ∑_ r=1 ^ 2013 (x)/(x^2+r^2) ) ( ∏_ r=1 ^ 2013 ( x^2+r²) ) dx = (1)/(2) [ ∏_ r=1 ^ 2013 ( 1+r²) - k² ] then k =

Black Book Advanced Problems in Mathematics Solution View → 127

f(x) = 2x - ^ -1 x - (x + 1+x^2 )

Black Book Advanced Problems in Mathematics Solution View → 128

The value of the definite integral ∫_0^ π/2 dx x + x + cosec x + x is :

Black Book Advanced Problems in Mathematics Solution View → 129

The value of the definite integral ∫_3^7 ( x^2)/( x^2 + (10-x)^2) dx is :

Black Book Advanced Problems in Mathematics Solution View → 130

The value of the integral ∫_ e^ -1 ^ e^2 | ( x)/(x) | dx is :

Black Book Advanced Problems in Mathematics Solution View → 131

The value of _ x → (π)/(4) ∫_2^ cosec^2 x t g(t) dt x^2 - (π^2)/(16) is :

Black Book Advanced Problems in Mathematics Solution View → 132

The value of _ n → ∞ ∑_ k=1 ^n (n-k)/(n^2) (4k)/(n) equals :

Black Book Advanced Problems in Mathematics Solution View → 133

For each positive integer n , define a function f_n on as follows : f_n(x) = cases 0 & if x = 0 (π)/(2n) & if 0 < x ≤ (1)/(n) (2π)/(2n) & if (1)/(n) < x ≤ (2)/(n) (3π)/(2n) & if

Black Book Advanced Problems in Mathematics Solution View → 134

Let n be a positive integer, then ∫_0^ n+1 \ |x-1|, |x-2|, |x-3|, … , |x-n| \ dx equals

Black Book Advanced Problems in Mathematics Solution View → 135

For positive integers k = 1, 2, 3 … , n , let S_k denotes the area of Δ AOB_k (where 'O' is origin) such that ∠ AOB_k = (kπ)/(2n) , OA = 1 and OB_k = k . The value of the _ n → ∞

Black Book Advanced Problems in Mathematics Solution View → 136

If A = ∫_0^1 ∏_ r=1 ^ 2014 (r-x) dx and B = ∫_0^1 ∏_ r=0 ^ 2013 (r+x) dx , then :

Black Book Advanced Problems in Mathematics Solution View → 137

If f(x) = [ (x)/(120) + (x^3)/(30) ] defined in , then ∫_0^1 (f(x) + 2) dx = (where [·] denotes greatest integer function)

Black Book Advanced Problems in Mathematics Solution View → 138

If f(x) = ∫_0^ g(x) dt 1+t^3 , g(x) = ∫_0^ x (1+ t)^2 dt , then the value of f' ((π)/(2))is equal to :

Black Book Advanced Problems in Mathematics Solution View → 139

Let f(x) = (1)/(x^2) ∫_0^x (4t^2 - 2f'(t)) dt , find 9f'(4)

Black Book Advanced Problems in Mathematics Solution View → 140

Evaluate _ n → ∞ ( (1^2)/(n^3+1^3) + (2^2)/(n^3+2^3) + (3^2)/(n^3+3^3) + … … + (4)/(9n) )

Black Book Advanced Problems in Mathematics Solution View → 141

The value of ∫_0^ 2π ^ -1 ( (1 - ^2 (x)/(2))/(1 + ^2 (x)/(2)) ) dx is :

Black Book Advanced Problems in Mathematics Solution View → 142

Given a function ' g ' continuous everywhere such that ∫_0^1 g(t) dt = 2 and g(1) = 5 . If f(x) = (1)/(2) ∫_0^x (x-t)^2 g(t) dt , then the value of f'''(1) - f''(1) is :

Black Book Advanced Problems in Mathematics Solution View → 143

If ∫_0^π (x^3 ^4 x ^2 x)/(π^2 - 3π x + 3x^2) dx = λ ∫_0^ π/2 ^2 x dx , then the value of λ is :

Black Book Advanced Problems in Mathematics Solution View → 144

∫_0^ 3 ( (1)/(2) (d)/(dx) ( ^ -1 (2x)/(1-x^2) ) ) dx equals to :

Black Book Advanced Problems in Mathematics Solution View → 145

Let y = \ x\ ^ [x] then the value of ∫_0^3 y dx equals to : *(where \ ·\ and [·] denote fractional part and greatest integer function respectively.)*

Black Book Advanced Problems in Mathematics Solution View → 146

The value of ∫_0^ 4/π ( 3x^2 (1)/(x) - x (1)/(x) ) dx is :

Black Book Advanced Problems in Mathematics Solution View → 147

The number of values of x satisfying the equation : ∫_ -1 ^x ( 8t^2 + (28t)/(3) + 4 ) dt = (3)/(2)x + 1 _ (x+1) x+1 , is :

Black Book Advanced Problems in Mathematics Solution View → 148

_ n → ∞ (1+2^4+3^4+…+n^4)/(n^5) - _ n → ∞ (1+2^3+3^3+…+n^3)/(n^5) is :

Black Book Advanced Problems in Mathematics Solution View → 149

The value of _ x → 0^+ ∫_1^ x ( ^ -1 t) dt 2x - 2x is equal to :

Black Book Advanced Problems in Mathematics Solution View → 150

Consider a parabola y = (x^2)/(4) and the point F (0, 1) . Let A_1(x_1, y_1), A_2(x_2, y_2), A_3(x_3, y_3), … , A_n(x_n, y_n) are ' n ' points on the parabola such x_k > 0 and ∠

Black Book Advanced Problems in Mathematics Solution View → 151

The minimum value of f(x) = ∫_0^3 e^ |x-t| dt where x ∈ is :

Black Book Advanced Problems in Mathematics Solution View → 152

If ∫_0^∞ ( x)/(x) dx = (π)/(2) , then ∫_0^∞ ( ^3 x)/(x) dx is equals to :

Black Book Advanced Problems in Mathematics Solution View → 153

∫ 1+ x ( (x)/(2) - (x)/(2) ) dx =

Black Book Advanced Problems in Mathematics Solution View → 154

If I_n = ∫_0^π ( (2nx))/( 2x) dx , then the value of I_ n+(1)/(2) is equal to (n ∈ I) :

Black Book Advanced Problems in Mathematics Solution View → 155

The value of function f(x) = 1 + x + ∫_1^x ( ^2 t + 2 t) dt where f'(x) vanishes is :

Black Book Advanced Problems in Mathematics Solution View → 156

Let f be a differentiable function on R and satisfies f(x) = x^2 + ∫_0^x e^ -t f(x-t) dt ; then ∫_0^1 f(x) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View → 157

The value of the definite integral ∫_ -π/2 ^ π/2 ( ^2 x)/(1 + 5^x) dx equals to :

Black Book Advanced Problems in Mathematics Solution View → 158

∫ ( (x^2 - x + 1)/(x^2 + 1) ) e^ ^ -1 (x) dx = f(x) · e^ ^ -1 (x) + C where C is constant of integration. Then f(x) is equal to :

Black Book Advanced Problems in Mathematics Solution View → 159

_ n → ∞ (1)/(n^3) ( n^2+1 + 2 n^2+2^2 + … + n n^2+n^2 ) = :

Black Book Advanced Problems in Mathematics Solution View → 160

∫ ((x^3 - 1))/((x^4 + 1)(x + 1)) dx , is :

Black Book Advanced Problems in Mathematics Solution View → 161

The value of _ x → 0^+ ∫_1^ x ( ^ -1 t) dt 2x - 2x is equal to :

Black Book Advanced Problems in Mathematics Solution View → 162

Let f(x) = _ n → ∞ x 1 + ( ^ -1 x)^n , then ∫_0^∞ f(x) dx =

Black Book Advanced Problems in Mathematics Solution View → 163

The value of _ n → ∞ ∑_ k=1 ^n ( (k)/(n^2 + n + 2k) ) =

Black Book Advanced Problems in Mathematics Solution View → 164

The value of _ y → 1^+ (∫_1^y |t - 1| dt)/( (y - 1)) is :

Black Book Advanced Problems in Mathematics Solution View → 165

Given that ∫ (dx)/((1 + x^2)^n) = x 2(n - 1)(1 + x^2)^ n-1 + ((2n - 3))/(2(n - 1)) ∫ dx (1 + x^2)^ n-1 . Find the value of ∫_0^1 (dx)/((1 + x^2)^4) : (you may or may not use

Black Book Advanced Problems in Mathematics Solution View → 166

Find the value of ∫_0^ π/4 ( x)^4 dx :

Black Book Advanced Problems in Mathematics Solution View → 167

∫ ( 9x + 6x)/(2 5x - 1) dx = A 4x + B x + C , then A + B is equal to : (Where C is constant of integration)

Black Book Advanced Problems in Mathematics Solution View → 168

∫ dx x^ 2014 + x = (1)/(p) ( (x^q)/(1 + x^r) ) + C where p, q, r ∈ N then the value of (p + q + r) equals (Where C is constant of integration)

Black Book Advanced Problems in Mathematics Solution View → 169

If ∫_0^1 e^ -x^2 dx = a , then ∫_0^1 x^2 e^ -x^2 dx is equal to

Black Book Advanced Problems in Mathematics Solution View → 170

If f(x) is a continuous function for all real values of x and satisfies ∫_0^n f(x) dx = (n^2)/(2) \; ∀ n ∈ I , then ∫_ -3 ^5 f(|x|) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View → 171

If ∫ (dx)/(x^4 (1 + x^3)^2) = a | (1 + x^3)/(x^3) | + (b)/(x^3) + (c)/(1 + x^3) + d , then (where d is arbitrary constant)

Black Book Advanced Problems in Mathematics Solution View → 172

_ n → ∞ ( 1 n n+1 + 1 n n+2 + … + 1 n 4n ) is equal to :

Black Book Advanced Problems in Mathematics Solution View → 173

Let f(x) = ∫_0^x dy 1 + y^3 . The value of the integral ∫_0^2 x f(x) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View → 174

The value of the definite integral ∫_0^ π/3 (1 + 3 x) dx equals

Black Book Advanced Problems in Mathematics Solution View → 175

If ∫_0^ 100 f(x) dx = a , then ∑_ r=1 ^ 100 ∫_0^1 f(r - 1 + x) dx =

Black Book Advanced Problems in Mathematics Solution View → 176

The value of ∫_0^1 ( _ n → ∞ ∑_ k=0 ^n x^ k+2 2^k k! ) dx is :

Black Book Advanced Problems in Mathematics Solution View → 177

Evaluate : ∫ x^5 1 + x^3 dx .

Black Book Advanced Problems in Mathematics Solution View → 178

If f(x) = ∫_0^x ( t)/(t) dt , which of the following is true ?

Black Book Advanced Problems in Mathematics Solution View → 179

Evaluate : ∫ (x^3 + 3x^2 + x + 9)/((x^2 + 1)(x^2 + 3)) dx .

Black Book Advanced Problems in Mathematics Solution View → 180

∫ ^5 x ^3 x dx equals to :

Black Book Advanced Problems in Mathematics Solution View → 181

_ x → 0 ∫_0^x e^ (tx) dt x equals to :

Black Book Advanced Problems in Mathematics Solution View → 182

If A = ∫_0^π ( x)/(x) dx , then ∫_0^ π/2 ( 2x)/(x) dx is equal to :

Black Book Advanced Problems in Mathematics Solution View →
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Related JEE Mathematics chapters

Application of Derivatives (795)Three Dimensional Geometry (756)Sequences and Series (748)Functions (700)Definite Integration (692)Vector Algebra (616)Differential Equations (586)Continuity and Differentiability (576)Probability (561)Complex Number (551)Binomial Theorem (550)Quadratic Equation (517)