1. FTC Review; The Method of Substitution
February 4, 2004

2. The Definite Integral as Area
Let f be a continuous function defined on the interval [a, b]. The definite integral of f from a to b, denoted by
represents the total signed area of the region bounded by y = f (x), the vertical lines x = a and x = b, and the x-axis.

3. Properties of Definite Integrals
Let f and g be continuous functions defined on the interval [a, b]. Furthermore, let c and k be constants such that a < c < b. Then…

4. The Fundamental Theorem of Calculus
Let f be a continuous function defined on [a, b], and let F be any antiderivative of f. Then

5. Keeping It Straight Definite Integral Area Function
Represents a real number (a signed area).
Area Function
Represents a single antiderivative of f.
Indefinite Integral
Represents the entire family of antiderivatives of f.

6. Substitution Rule for Indefinite Integrals

7. Implementing the Substitution Rule
Choose u.
Differentiate u w.r.t. x and solve for du.
Substitute u and du into the old integral involving x to form a new integral involving only u.
Antidifferentiate with respect to u.
Re-substitute to find the antiderivative as a function of x.

8. Two Special Forms

9. Substitution Rule for Definite Integrals

10. Implementing the Substitution Rule (Definite Integrals)
Choose u = g(x).
Differentiate u w.r.t. x and solve for du.
Substitute u and du into the old integral involving x, as well as converting endpoints from a and b to g(a) and g(b).
Antidifferentiate with respect to u and evaluate at the new endpoints.

12. Arcsine (Inverse Sine Function)
For x in [-1, 1], y = arcsin x is defined by the conditions
x = sin y and
–/2  y  /2.
In words, arcsin x is the angle between –/2 and /2 whose sine is x.