Calculus - Antiderivatives and Integrals

Antiderivatives and Integrals

Antiderivative

F is an antiderivative of f if F’(x) = f(x).

General form: If F is antiderivative, then F(x) + C is family of all antiderivatives.

Indefinite Integral

$\int f(x)dx = F(x) + C$

Definite Integral (Riemann Sum)

$\int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x$

where Δx = (b-a)/n.

Fundamental Theorems of Calculus

FTC Part 1: If f is continuous on [a,b], then function F(x) = ∫[a to x] f(t)dt is differentiable on (a,b) and F’(x) = f(x).

FTC Part 2: If f is continuous on [a,b] and F is antiderivative of f, then: $\int_a^b f(x)dx = F(b) - F(a)$

Basic Integration Formulas

• ∫ dx = x + C
• ∫ xⁿ dx = x^(n+1)/(n+1) + C (n ≠ -1)
• ∫ 1/x dx = ln|x| + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = a^x/ln a + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C
• ∫ csc² x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ csc x cot x dx = -csc x + C

Integration Rules

Constant Multiple: ∫ cf(x)dx = c∫f(x)dx

Sum/Difference: ∫ [f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

Substitution: ∫ f(g(x))g’(x)dx = ∫ f(u)du where u = g(x)

Average Value

$\text{Avg}(f) = \frac{1}{b-a}\int_a^b f(x)dx$