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Indefinite and Definite Integration — JEE Main & Advanced Mathematics PYQs

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

182 questionsMathematicsSolutions on every page
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 ( , 0) If f : R - (2n + 1) 2 R then f(x) be a :

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**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 ( , 0) The number of solution(s) of the equation f(x) = x^3 in

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**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 minimum nu

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**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 f'(1 +

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**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 - t) e

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**Paragraph for Question Nos. 6 to 8** Consider the function f(x) and g(x) , both defined from R Rf(x) = x^3 2 + 1 - x ₀^x g(t) dt and g(x) = x - ₀^1 f(t) dt , then Minimum value o

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**Paragraph for Question Nos. 6 to 8** Consider the function f(x) and g(x) , both defined from R Rf(x) = x^3 2 + 1 - x ₀^x g(t) dt and g(x) = x - ₀^1 f(t) dt , then The number of p

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**Paragraph for Question Nos. 6 to 8** Consider the function f(x) and g(x) , both defined from R Rf(x) = x^3 2 + 1 - x ₀^x g(t) dt and g(x) = x - ₀^1 f(t) dt , then The area bounde

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**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 constant. b

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**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 constant. T

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**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 constant. ₀

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**Paragraph for Question Nos. 12 to 13** Let f₀(x) = x and for n 0 and x > 0 Let f_ n+1 (x) = ₀^x f_n(t) dt then : f₃(x) equals :

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**Paragraph for Question Nos. 12 to 13** Let f₀(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)

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

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

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**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 value of C is

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**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 value of L is

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Match the following columns: array |l|l| 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 ( 1 x - 1 ) dx = & (Q) 1 (C)

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Match the following f(x) dx is equal to, if array |l|l| 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

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Match the following columns: array |l|l| Column-I & Column-II (A) ₀^ /2 x (1 + x)(2 + x) dx = & (P) 6 (B) ₀^ 41 /4 | x| dx = & (Q) 20 + 1 2 (C) _ -1/2 ^ 1/2 [ [x] + ( 1 + x 1 - x )

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Match the following columns: array |l|l| 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 =

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Match the following columns: array |l|l| Column-I & Column-II (A) ₀^ 1.5 [x^2] dx & (P) - (B) ₀^4 x dx (where x denotes the fractional part of x ) & (Q) 4( 2 - 1) (C) ₀^ 2 [ x + x]

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x + ( 3x)^2 1 - 9x^2 dx = 1 k₁ ( 1 - 9x^2 + ( ⁻¹ 3x)^ k₂ ) + C , then k₁^2 + k₂^2 = *(where C is an arbitrary constant.)*

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If ₀^ x^3 (a^2 + x^2)^5 dx = 1 k a^6 , then find the value of k 8 .

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Let f(x) = x x ; x [ 3 2 , 2 ] and g(x) be its inverse. If ₀^ 2 g(x) dx = ^2 + + , where , and R , then find the value of 2( + + ) .

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

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

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

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Let ₀^1 4x^3(1 + (x^4)²⁰¹⁰) (1 + x^4)²⁰¹² dx = where and are relatively prime positive integers. Find unit digit of .

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

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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 .

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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 = ^a is constant (where is constant.)

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Let I = ₀^ x^6 ( - x)^8 dx , then ¹⁵ 15 9 I =

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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 .

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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) .

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If x denotes the fractional part of x , then I = ₀¹⁰⁰ x dx , then the value of 9I 155 is:

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Let I_n = ₀^ ( (n + 1 2 )x ) ( x 2 ) dx where n W . If I₁^2 + I₂^2 + I₃^2 + + I₂₀^2 = m ^2 , then find the largest prime factor of m .

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If M be the maximum value of 72 ₀^y x^4 + (y - y^2)^2 dx for y , then find M 6 .

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

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

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

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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) .

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If f(n) = 1 ₀^ /2 ^2(n ) d ^2 , n N , then evaluate f(15) + f(3) f(12) - f(10) .

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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 ₀⁵⁰ f(x) dx = I . Find [ I - 3] . *(where [ ] denotes greatest integer function.)*

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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 pq(p+q) 10 .

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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| .

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If _a^b | x| dx = 8 and ₀^ a+b | x| dx = 9 then the value of 1 2 _a^b x x dx is:

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If f(x), g(x), h(x) and (x) are polynomial in x , ( ₁^x f(x) h(x) dx ) ( ₁^x g(x) (x) dx ) - ( ₁^x f(x) (x) dx ) ( ₁^x g(x) h(x) dx ) is divisible by (x - 1)^ . Find maximum value

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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) .

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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 .

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For a positive integer n , let I_n = _ - ^ ( 2 - |x| ) nx dx . Find the value of [I₁ + I₂ + I₃ + I₄] where [ ] denotes greatest integer function. ***

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dx (1 + x )^8 = - 1 3(1 + x )^ k₁ + 2 7(1 + x )^ k₂ + C , then :

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

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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 :

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

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

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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 :

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If f( ) = _ n _ r=0 ^ n 2r n (3 n - 2r)(n + 2r) then :

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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 :

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

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

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Let l₁ = _ x x - ^2 x x + x and l₂ = _ h 0^+ _ -1 ^1 h dx h^2 + x^2 . Then :

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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 :

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

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

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Let L = _ n _a^ n dx 1 + n^2 x^2 where a R then L can be :

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Let I = ₀^1 1 + x 1 - x dx and J = ₀^1 1 - x 1 + x dx then correct statement(s) is/are :

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a^x ( x + a ( x e )^x ) dx =

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

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

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The value of the integral ₀^2 (x^2 + 2) (x + 2)^2 dx is :

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If I₁ = ₀^1 1 + x^8 1 + x^4 dx and I₂ = ₀^1 1 + x^9 1 + x^3 dx , then :

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Let f:(0, 1) (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 - (f(s))^2 ds f(t) -

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If f( ) = 4 3 (1 - ^6 - ^6 ) , then _ n 1 n [ f ( 1 n ) + f ( 2 n ) + f ( 3 n ) + + f ( n n ) ] =

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The value of ₀^1 (x^6 - x^3) (2x^3 + 1)^3 dx is equal to :

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2 ₀^ 1 2 ⁻¹ x x dx - ₀^1 ⁻¹ x x dx =

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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 =

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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.)*

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For any x R , and f be a continuous function. Let I₁ = _ ^2 x ^ 1+ ^2 x t f(t(2-t))dt , I₂ = _ ^2 x ^ 1+ ^2 x f(t(2-t))dt , then I₁ =

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

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

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

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dx 1 - ^2 x = 1 ⁻¹( x) + C , then =

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

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If I_n = ( x)^n dx; n N , then 5I₄ - 6I₆ is equal to:

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x^2 (a + bx)^2 dx equals to :

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8x⁴³ + 13x³⁸ (x¹³ + x^5 + 1)^4 dx =

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

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(1 + x - x⁻¹) e^ x + x⁻¹ dx =

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

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

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The value of the definite integral ₀^1 1 + x + x+x^2 x + 1 + x dx is :

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x^ x^2+1 (2 x + 1) dx

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

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

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

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

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

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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 =

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(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.)*

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

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[ 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) = ^2 ₀^1 t (x + t) dt over all real number x :

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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 ' is:

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 :

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Let a function f: R R be defined as f(x) = x + x . The value of ₀^ 2 f⁻¹(x) dx will be :

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

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

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If f(x) = x 1 + ( x)( x) ; x [1, ) then ₁^ 2e f(x) dx is:

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₀^2 (y^2 - 4y + 5) (y - 2) (2y^2 - 8y + 11) dy is equal to :

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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^ +h x^2 e^ -x^2 dx - ₀^ x^2 e^ -x^2 dx h is equal to :

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Let f: R^+ 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^ /2 e^ x - [ x] d( ^2 x - [ ^2 x]) is , then [ - 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

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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 :

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