PrepWiser › Learn › Mathematics › Definite IntegrationDefinite Integration — JEE Main PYQs & Practice Questions Definite Integration is a Mathematics chapter in the JEE Main syllabus. This page brings together 223 previous-year and practice questions on Definite Integration, organised into 4 topics: Properties of Definite Integrals, Fundamental Theorem of Calculus, Leibniz Rule and Definite Integral as Limit of Sum. Most questions come with a step-by-step solution you can work through free in PrepWiser.
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Properties of Definite Integrals Let the function f:[0,2] → ℝ be defined as f(x)= e^(min {x², x-[x]},) x ∈[0,1)… If for all real triplets (a, b, c), ƒ(x) = a + bx + cx²; then ∫₀¹ f(x)dx is equal to : The value of ∫₀¹(2 x³-3 x²-x+1)^(1/3) d x is equal to : Let y=f(x) be a thrice differentiable function in (-5,5). Let the tangents to the curve… Let g(x) = ∫₀ˣ f(t)dt, where f is continuous function in [ 0, 3 ] such that 1/3 ≤ f(t) ≤… The value of ∫_((- 1)/√2)^(1/√2) ( ( (x + 1)/(x - 1) )² + ( (x - 1)/(x + 1) )² - 2… If α=∫₀^(2√3)log₂(x²+4) dx+∫₂⁴√(2ˣ-4) dx, then α² is equal to ________. If f(α)=∫₁^α (log ₁₀ t)/(1+t) dt, α>0, then f(e³)+f(e⁻³) is equal to : The function f(x), that satisfies the condition f(x) = x + ∫₀^(π /2) sin x.cos y f(y)… Let f(x) = x + a/(π ² - 4)sin x + b/(π ² - 4)cos x,x ∈ R be a function which satisfies… Let Iₙ(x)=∫₀ˣ 1/(t²+5)ⁿ d t, n=1,2,3, … . Then : The integral ∫_(π/6)^(π/3) tan ³x.sin ²3x( 2sec ²x.sin ²3x + 3 tan x.sin 6x )dx is equal… ∫0^(π / 4) (cos ² x sin ² x)/(cos ³ x+sin ³ x)² d x is equal to If ∫-0.15^(0.15)|100 x²-1| d x=k/3000, then k is equal to ___________. The value of the integral ∫₀^(π/2) 60 (sin (6 x))/(sin x) d x is equal to _________. ∫₆¹⁶ (log ₑx²)/(log ₑx² + log ₑ(x² - 44x + 484))dx is equal to : Let f : R → R be a continuous function satisfying f(x) + f(x + k) = n, for all x ∈ R… Let [t] denote the greatest integer function. If ∫0^(2.4)[x²] d x=α+β √2+γ √3+δ √5, then… If α=1 and β=1+i√2, where i=√-1 are two roots of the equation x³+ax²+bx+c=0,a,b,∈ ℝ… The value of the integral ∫₀^π |sin 2x|dx is ___________. Fundamental Theorem of Calculus If ∫₀¹ 1/(√(3+x)+√(1+x)) d x=a+b √2+c √3, where a, b, c are rational numbers, then 2 a+3… Let f be a continuous function satisfying ∫0^(t²)(f(x)+x²) d x=4/3 t³, ∀ t > 0. Then… The value of ∫π/3^(π/2) (2+3 sin x)/(sin x(1+cos x)) d x is equal to : Let ∫α^(log ₑ 4) (d x)/√(eˣ-1)=π/6. Then e^α and e^(-α) are the roots of the equation : If ∫₀^(√3) (15 x³)/√(1+x²+√(1+x²)³) d x=α √2+β √3, where α, β are integers, then α+β is… Let aₙ = ∫₋₁ⁿ ( 1 + x/2 + x²/3 + ..... + xⁿ⁻¹/n )dx for every n ∈ N. Then the sum of all… Let for some function y=f(x), ∫₀ˣ t f(t) d t=x² f(x), x>0 and f(2)=3. Then f(6) is equal… The integral ∫₁² eˣ.xˣ( 2 + log ₑx ) dx equals : Let f: ℝ → ℝ be a function defined by f(x)=x/(1+x⁴)^(1 / 4), and g(x)=f(f(f(f(x))))… Let f : R → R be defined as f(x) = e⁻ˣsinx. If F : [0, 1] → R is a differentiable… Let α>0. If ∫₀^α x/(√(x+α)-√x) d x=(16+20 √2)/15, then α is equal to : The value of b > 3 for which 12∫₃ᵇ 1/(x² - 1)(x² - 4)dx = log ₑ( 49/40 ) , is equal to… Let f(x)=∫₀ˣ t(t²-9 t+20) dt, 1 ≤ x ≤ 5. If the range of f is [α, β], then 4(α+β) equals : Let f:(0, ∞) → R and F(x)=∫₀ˣ t f(t) dt. If F(x²)=x⁴+x⁵, then Σᵣ₌₁¹² f(r²) is equal to… The integral 16∫₁² dx/x³( x² + 2 )² is equal to Let f:(0, ∞) → R be a twice differentiable function. If for some a≠ 0, ∫₀¹ f(λ x) d λ=a… If [t] denotes the greatest integer ≤ t, then the value of 3(e - 1)/e∫₁² x²e^([x] +… Let f:[1, ∞) →[2, ∞) be a differentiable function. If 10 ∫₁ˣ f(t) dt=5 x f(x)-x⁵-9 for… Let f(x)=x/(1+xⁿ)^(1/n), x ∈ ℝ-{-1}, n ∈ ℕ, n > 2. If fⁿ(x)=(f ° f ° f … .. upto n… Let f: ℝ → ℝ be defined as f(x)=a e²ˣ+b eˣ+c x. If f(0)=-1, f′(log ₑ 2)=21 and ∫₀^(log ₑ… Leibniz Rule Let f:[1,∞)→ ℝ be a differentiable function. If 6∫₁ˣf(t)dt=3xf(x)+x³-4 for all x≥ 1… Let f be a differentiable function satisfying f(x)=2/√3 ∫₀^(√3) f((λ² x)/3) d λ, x>0 and… If φ(x)=1/√x ∫_(π/4)ˣ(4 √2 sin t-3 φ′(t)) d t, x>0, then ∅′(π/4) is equal to : Let f be a twice differentiable function on ℝ. If f′(0)=4 and f(x) + ∫₀ˣ (x - t)f'(t)dt… Let f be a differentiable function in ( 0,π/2 ). If ∫_(cos x)¹ t² f(t)dt = sin ³x + cos… Let F:[3,5] → R be a twice differentiable function on (3, 5) such that F(x) = e⁻ˣ∫₃ˣ… The number of elements in the set S={x:x∈ [0,100] and ∫₀ˣt²sin (x-t)dt=x²} is ____ Let f be a real valued continuous function defined on the positive real axis such that… Let a differentiable function f satisfy f(x)+∫3ˣ f(t)/t d t=√(x+1), x ≥ 3. Then 12 f(8)… Let ∫0ˣ √(1-(y′(t))²) d t=∫₀ˣ y(t) d t, 0 ≤ x ≤ 3, y ≥ 0, y(0)=0. Then at x=2, y^(′… Let f be a differentiable function satisfying f(x)=1-2x+∫₀ˣe⁽ˣ⁻ᵗ⁾f(t) dt, x∈ ℝ and let… Let A=[1, 3, -1; 2, 1, α; 0, 1, -1] be a singular matrix. Let f(x)=∫₀ˣ(t²+2t+3)dt,x∈… Let f(x) = ∫₀ˣ eᵗf(t)dt + eˣ be a differentiable function for all x∈R. Then f(x) equals : Definite Integral as Limit of Sum More Mathematics chapters Frequently asked questions How many Definite Integration questions are there for JEE Main? PrepWiser has 223 Definite Integration questions for JEE Main, grouped into 4 topics such as Properties of Definite Integrals, Fundamental Theorem of Calculus and Leibniz Rule.
Are these Definite Integration questions from previous-year JEE Main papers? Yes — most are genuine previous-year questions from past JEE Main papers, and the rest are practice questions written to match the JEE Main pattern and difficulty.
Where can I see the answers and solutions to these Definite Integration questions? Open any Definite Integration question in PrepWiser (it is free) to attempt it; most come with a step-by-step solution.