Do I need + C in a definite integral?

No. The constant cancels when you subtract F(a) from F(b), so a definite integral is a number with no constant of integration.

How do I know if a function is odd or even?

Replace x by −x. If f(−x) = f(x) it is even; if f(−x) = −f(x) it is odd. Products follow sign rules: odd × odd is even, odd × even is odd.

Can a definite integral be negative?

Yes. Where the curve lies below the x-axis the integral counts negatively. That is why area questions need the region split at the x-intercepts.

Definite Integration (DSE M2): Properties, Substitution and Odd / Even Functions

A definite integral is the limit of a sum, evaluated by the Fundamental Theorem of Calculus: ∫_a^b f(x) dx = F(b) − F(a). When you substitute u = g(x), change the limits to g(a) and g(b). Over [−a, a] an odd function integrates to 0 and an even function to twice its integral over [0, a].

Solve Definite Integration questions with AI

y = x³ on [−2, 2]: the region below the axis on the left cancels the region above it on the right, so the integral of an odd function over [−a, a] is 0. Try it yourself →

Properties of Definite Integrals

Fundamental Theorem∫_a^b f(x) dx = F(b) − F(a)

Reversing the limits∫_a^b f(x) dx = −∫_b^a f(x) dx

Splitting the interval∫_a^b = ∫_a^c + ∫_c^b

Odd function∫_−a^a f(x) dx = 0

Even function∫_−a^a f(x) dx = 2∫₀^a f(x) dx

By parts∫_a^b u dv = [uv]_a^b − ∫_a^b v du

Worked Examples — Definite Integration

Example 1: Using odd and even functions

Evaluate ∫₋₂² (x³ cos x + x²) dx.

x³ cos x is odd (odd × even), so its integral over [−2, 2] is 0
x² is even, so its integral is 2∫₀² x² dx
2 × [x³/3]₀² = 2 × 8/3 = 16/3

Answer: 16/3

Example 2: Substitution with a change of limits

Evaluate ∫₀¹ x / (x² + 1) dx.

Let u = x² + 1, du = 2x dx
Limits: x = 0 → u = 1, x = 1 → u = 2
∫₁² ½ · (1/u) du = ½ [ln u]₁²
= ½ ln 2

Answer: ½ ln 2

💡 Exam Traps & Secrets

⚠️ Old limits with a new variableAfter substituting u = g(x), the limits must be values of u. Keeping the x-limits while integrating in u gives a wrong number even when the antiderivative is right.

⚠️ Check symmetry before integratingWhen the limits are −a and a, test each term for odd or even first. Examiners often include a term that looks impossible to integrate but is odd, so it contributes 0.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(By parts with limits)Evaluate the integral of x cos x from 0 to π/2.

(Substitution with limits)Evaluate the integral of x √(1 − x²) from 0 to 1.

(Odd and even functions)Evaluate the integral of (x⁵ + sin x + 3) from −1 to 1.