Module 6: Integration
Calculus I – Study Guide
Learn Without Walls
1. Antiderivatives and Indefinite Integrals
An antiderivative of f(x) is a function F(x) such that F'(x) = f(x). The general antiderivative is F(x) + C.
Basic Antiderivative Rules
| Function f(x) | Antiderivative F(x) + C |
|---|---|
| xn (n ≠ −1) | xn+1/(n+1) |
| 1/x | ln|x| |
| ex | ex |
| sin x | −cos x |
| cos x | sin x |
| sec² x | tan x |
Always include + C for indefinite integrals!
2. Riemann Sums
Δx = (b − a)/n. Left sum uses left endpoints; right sum uses right endpoints; midpoint sum uses midpoints of each subinterval.
Rn = ∑i=1n f(xi) Δx
As n → ∞, Riemann sums converge to the definite integral (for integrable functions).
3. The Definite Integral
∫ab f(x) dx = limn→∞ ∑ f(xi*) Δx
Key Properties
∫aa f(x) dx = 0
∫ba f(x) dx = −∫ab f(x) dx
∫ab [cf(x)] dx = c ∫ab f(x) dx
∫ab [f ± g] dx = ∫ab f dx ± ∫ab g dx
∫ac f dx = ∫ab f dx + ∫bc f dx
4. Fundamental Theorem of Calculus
Part 1 (Evaluation Theorem)
∫ab f(x) dx = F(b) − F(a), where F' = f
Example: ∫13 x² dx = [x³/3]13 = 9 − 1/3 = 26/3.
Part 2 (Derivative of an Integral)
d/dx [∫ax f(t) dt] = f(x)
d/dx [∫au(x) f(t) dt] = f(u(x)) · u'(x) [chain rule version]
Example: d/dx [∫0x² sin t dt] = sin(x²) · 2x.
5. Net Change Theorem
∫ab F'(x) dx = F(b) − F(a)
The integral of a rate of change = net change in the quantity.
Displacement = ∫ab v(t) dt (can be negative)
Total distance = ∫ab |v(t)| dt (always non-negative)
Strategy for total distance: Find where v(t) = 0, split the integral at those points, take the absolute value of each piece, and add.