Module 6 Quick Reference

Integration – Calculus I

Basic Antiderivative Rules

Power Rule

∫ xn dx = xn+1/(n+1) + C, n ≠ −1

Special Cases

∫ 1/x dx = ln|x| + C

∫ ex dx = ex + C

Trig

∫ sin x dx = −cos x + C

∫ cos x dx = sin x + C

More Trig

∫ sec² x dx = tan x + C

∫ csc² x dx = −cot x + C

Definite Integral Properties

∫aa f dx = 0  |  ∫ba f dx = −∫ab f dx  |  ∫ac f dx = ∫ab f dx + ∫bc f dx
∫ab [cf(x)] dx = c∫ab f dx  |  ∫ab [f ± g] dx = ∫ab f dx ± ∫ab g dx

Fundamental Theorem of Calculus

Part 1 (Evaluation)

∫ab f(x) dx = F(b) − F(a)

where F' = f

Part 2 (Derivative)

d/dx [∫ax f(t) dt] = f(x)

Chain rule: f(u(x)) · u'(x)

FTC connects differentiation and integration as inverse operations.

Net Change Theorem

∫ab F'(x) dx = F(b) − F(a)  (integral of rate = net change)

Displacement

∫ab v(t) dt

Can be negative (signed)

Total Distance

∫ab |v(t)| dt

Always non-negative

Riemann Sum Formulas

Δx = (b − a)/n  |  Rn = ∑ f(xi*) Δx  |  As n → ∞, Rn → ∫ab f(x) dx