Mathematics · JEE Main · JEE Advanced

Definite Integration: concepts, formulas & solved PYQ-style examples

Definite integration is a high-weightage chapter in JEE Maths. Many questions look impossible by direct integration and collapse in one line with the right property.

Before integrating, check symmetry: replace x with (a + b − x) and see whether the integrand simplifies.

Also for AP Calculus AB/BCAlso for A-level MathsAlso for IB Maths AA
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Key concepts

King’s property

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx — the most used trick.

Even and odd functions

Odd functions integrate to zero over symmetric limits.

Periodic functions

Integral over n periods equals n times one period.

Integration by parts

ILATE order to choose the first function.

Limit of a sum

Converting Σ (1/n) f(r/n) into ∫₀¹ f(x) dx.

Formula sheet

King’s property

∫abf(x) dx=∫abf(a+b−x) dx\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx

Even / odd

∫−aaf(x) dx={2∫0af(x) dxf even0f odd\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & f \text{ even} \\ 0 & f \text{ odd} \end{cases}

By parts

∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du

Limit as a sum

lim⁡n→∞1n∑r=1nf ⁣(rn)=∫01f(x) dx\lim_{n\to\infty} \dfrac{1}{n}\sum_{r=1}^{n} f\!\left(\dfrac{r}{n}\right) = \int_0^1 f(x)\,dx

What PYQs keep asking

Patterns that repeat across previous-year JEE and NEET papers for this chapter.

  • Integrals that simplify with King’s property, often with sin and cos or log terms.
  • Even/odd and periodic properties to cut the work in half.
  • Limits of sums converted into definite integrals.
  • Integrals involving the greatest integer or modulus function, split at break points.

Solved examples

Example 1

Evaluate I=∫0π/2sin⁡xsin⁡x+cos⁡x dx\displaystyle I = \int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx.

  1. 1King’s property with x→π2−xx \to \tfrac{\pi}{2} - x: I=∫0π/2cos⁡xcos⁡x+sin⁡x dxI = \int_0^{\pi/2} \dfrac{\cos x}{\cos x + \sin x}\,dx.
  2. 2Add the two forms: 2I=∫0π/21 dx=π22I = \int_0^{\pi/2} 1\,dx = \dfrac{\pi}{2}.
  3. 3So I=π4I = \dfrac{\pi}{4}.
Answer: π4\dfrac{\pi}{4}
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Example 2

Evaluate ∫01xex dx\displaystyle \int_0^1 x e^x\,dx.

  1. 1By parts with u=xu = x, dv=exdxdv = e^x dx: ∫xexdx=xex−ex\int x e^x dx = x e^x - e^x.
  2. 2Apply limits: [xex−ex]01=(e−e)−(0−1)[x e^x - e^x]_0^1 = (e - e) - (0 - 1).
  3. 3This simplifies to 11.
Answer: 1
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Common traps

  • Applying King’s property and forgetting to add the two forms of I.
  • Integrating |f(x)| or [x] without splitting at the points where they change.
  • Losing the minus sign when evaluating the lower limit.

Practise with

Derivative & Integral CalculatorProbability4 Integral Calculus practice questionsVideo solutions

FAQ

What is King’s property in definite integration?

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx. Adding the original and transformed integrals often cancels the hard part.

How do I recognise a limit of a sum?

Look for a sum with 1/n in front and terms like r/n; replace r/n with x and 1/n with dx.

Is definite integration important for JEE Main?

Yes, it is one of the most frequently tested calculus chapters in both JEE Main and Advanced.

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