Integration Flashcards
Antiderivative/Indefinite Integral
F is an antiderivative of f on an interval I, if F’(x) = f(x) for all x in I
∫f(x)dx= F(x) + C
f(x) is the Integrand
dx is the Variable of Integration
F(x) is the Antiderivative of x
C is the Constant of Integration
Integration is the opposite of…
Differentiation
Differentiation is the opposite of…
Integration
∫ (0) dx = ?
C
∫ (k) dx = ?
kx + C
∫ kf(x) dx = ?
k ∫ f(x) dx
∫ [f(x) ± g(x)] dx = ?
∫ f(x) dx ± ∫ g(x) dx
∫ x^n dx = ?
{[x^(n + 1)] / n + 1}, n ≠ -1
∫ cos(x) dx = ?
sin(x) + C
∫ sin(x) dx = ?
-cos(x) + C
∫ sec^2(x) dx = ?
tan(x) + C
∫ sec(x) tan(x) dx = ?
sec(x) + C
∫ csc^2(x) dx = ?
-cot(x) + C
∫ csc(x) cot(x) dx = ?
-csc(x) + C
Inscribed Rectangles
Rectangles that lie within/under the curve of a function
Circumscribed Rectangles
Rectangles that extend outside/go above the curve of a function
How to find Average Area under a Curve
Find the area under the curve and then divide by the range of the domain
Fundamental Theorem of Calculus
∫(b to a) f(x)dx = F(x) (b to a) = F(b) - F(a)
Mean Value Theorem
∫(b to a) f(x) dx = f(c)(b - a), there is a number c between [a,b] which is a rectangle whose area is equal to the area under the curve
Second Fundamental Theorem of Calculus
(d/dx) [ ∫(x to a) f(t) dt ] = f(x)
Net Change Theorem
∫(b to a) F’(x) dx = F(b) - F(a)
