Definite integrals

A definite integral has limits: ∫ₐᵇ f(x) dx. Its answer is a number, not a function.

How to evaluate it: find any anti derivative F, then work out F(b) – F(a): value at the top limit minus value at the bottom limit. We write this as [F(x)]ₐᵇ.

Example: ∫₀^(π/2) cos x dx = [sin x]₀^(π/2) = sin(π/2) – sin 0 = 1.

No C is needed: it would appear in both F(b) and F(a) and cancel.

This is the second fundamental theorem. The first says: if A(x) = ∫ₐˣ f(t) dt (an area that grows as x moves), then A′(x) = f(x). Example: if A(x) = ∫₀ˣ (t² + 1) dt, then A′(x) = x² + 1.

Substitution with limits: when you change x to t, change the limits to t values too. Then you never need to go back to x.

Example: ∫₀¹ 2x/(1 + x²)² dx. Put t = 1 + x², dt = 2x dx. When x = 0, t = 1; when x = 1, t = 2. The integral becomes ∫₁² dt/t² = [–1/t]₁² = –½ + 1 = ½.

Forgetting to change the limits is the most common mistake in this lesson.