Lakshya NDA

Paper I — Mathematics

Integration (Indefinite, Definite & Area)

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

Your study controls

Log in to mark complete, bookmark and save mastery.

Log in to track progress
High yield5 mapped NDA questions

Build the idea

Integration reverses differentiation and accumulates area. First look for a direct standard form, then substitution, partial fractions or integration by parts. For definite integrals, symmetry can remove most calculation.

tangent slope∫ area
Function, tangent and accumulated area

Concept

Integration is the reverse of differentiation and also measures area under a curve. Indefinite integrals include an arbitrary constant; definite integrals give a number using the limits.

Substitution and integration by parts (ILATE order) handle most cases.

Key Points & Formulas

  • ∫xⁿ dx = x^(n+1)/(n+1) + C (n≠−1); ∫(1/x)dx = ln|x| + C.
  • ∫sinx dx = −cosx + C; ∫eˣ dx = eˣ + C.
  • Definite: ∫ₐᵇ f(x)dx = F(b) − F(a).
  • Area under y=f(x) from a to b (above x-axis) = ∫ₐᵇ f(x) dx.

Worked Example

Q. Evaluate ∫₀² (3x² + 2) dx.

Antiderivative = x³ + 2x.

At x=2: 8 + 4 = 12; at x=0: 0.

Definite integral = 12 − 0 = 12.

Common traps

  • Omitting +C in an indefinite integral.
  • Changing variables but not changing definite limits.
  • Calling a signed integral geometric area when the curve crosses the axis.

NDA shortcuts

  • If the numerator resembles the derivative of the denominator, try substitution.
  • Use ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx.
  • For odd f, ∫₋ₐᵃ f(x)dx=0; for even f, it is twice ∫₀ᵃ.

One-screen revision sheet

Recall before you practise

  • ∫xⁿdx=xⁿ⁺¹/(n+1)+C
  • ∫u dv=uv−∫v du
  • ∫ₐᵇf=F(b)−F(a)

Past-paper bank

Practise every mapped question

5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA Practice 2026

∫01x dx\displaystyle\int_0^1 x\,dx equals

Question 2

NDA II 2025

Consider the following:

I. dudx=−v\dfrac{du}{dx} = -v

II. dvdx=−u\dfrac{dv}{dx} = -u

Which of the above is/are correct?

Question 3

NDA II 2024

What is the area of the parabola bounded by the latus rectum?

Question 4

NDA I 2024

What is the value of α?

Question 5

NDA II 2024

What is ∣U2(x)−V2(x)∣|U^2(x) - V^2(x)| equal to?

← Application of Derivatives (Tangents, Maxima & Minima)Continue in Test ModuleDifferential Equations →