
In physics we may know the velocity v(t) (the derivative) and wish to compute the position s(t) of an object. Since s'(t) = v(t), this amounts to finding a function whose derivative is v(t). A function F whose derivative is f is called an ________ of f. ________ will turn out to be the key to evaluating definite integrals
 antiderivative
 Antiderivative

Define Antiderivative
A function F is an antiderivative of f on an open interval (a,b) if F'(x) = f(x) for all x in (a,b)

F(x) = cosx is an antiderivative of f(x) = ______ because for all values of x,
F'(x) = ________ = ______ = f(x)
F(x) = 1/3x^{3} is an antideriative of f(x) = ______ because for all values of x,
F'(x) = _______ = ______ = f(x)
 sinx
 d/dx(cosx) = sinx
 x^{2}
 d/dx(1/3x^{3}) = x^{2}

One critical observation is that antiderivatives are not unique. We are free to add a constant C because the derivative of a constant is _____, and so, if F'(x) = _____, then (F(x) + C' =____. For example, each of the following is a antiderivative of x2:
1/3x^{3}, 1/3x^{3} + 5, 1/3x^{3}  6

Are there any antiderivatives of f other than those obtained by adding a constant to a given antiderivative F?
 No if f is defined on an open interval (a,b)
 **by The General Antiderivative Theorem

State the General Antiderivative Theorem
Let y = F(x) be an antiderivative of y = f(x) on (a,b). Then every other antiderivative on (a,b) is of the form y = F(x) + C for some constant C

Find two antiderivatives of f(x) = cosx. Then determine the general antiderivative
The functions F(x) = sinx and G(x) = sinx + 2 are both antiderivatives of f(x) = cosx. The general antiderivative is F(x) = sinx + C, where C is ay constant

The process of finding an antiderivative is called ______. We will see why in the next section, when we discuss the connection between antiderivatives and areas under ______ given by the Fundamental Theorem of Calculus. Anticipating this result, we begin using the integral sign ∫, the standard notation for _________
 integration
 curves
 antiderivatives

Indefinite Integral:
The notation ∫ f(x)dx = F(x) + C means that F'(x) = ____
We say that y = F(x) + C is the general _______ or ______ ______ of y = f(x)
 f(x)
 antiderivative or indefinite integral

The terms "antiderivative" and "indefinite integral" are used ________. In some textbooks, and antiderivative is called a _______ function
 interchangeably
 primitive function

Define Integrand
The symbol dx is a _______, it is a part of o=the integral notation and serves to indicate the _______ _______. The constant C is called the constant of _________.
 Integrand: The expression f(x) appearing in the integral sign
 differntial
 independent variable
 integration

Some indefinite integrals can be evaluated by ________ the familiar derivative formulas. For example, we obtain the indefinite integral of y = x^{n} by reversing the Power Rule for derivatives
reversing

State The Power Rule for Integrals Theorem
 ∫ x^{n}dx = x^{n+1}/n+1 + C
 for n ≠ 1

Prove the Power Rule for Integrals Theorem
 We just need to verify that F(x) = x^{n+1}/n + 1 is an antiderivative of f(x) = x^{n}
 F'(x) = d/dx(x^{n+1}/n+1 + C)
 F'(x) = 1/n + 1((n+1)x^{n}) = x^{n}

In words, the Power Rule for Integrals instructs to follow two steps in order to integrate a power of x. State the steps
 1) add one to the exponent
 2) divide by the new exponent

The Power Rule is not valid for n = 1. In fact, for n = 1, what meaningless result do we obtain? It turns out that the antideriative of f(x) = x^{1} is the ______ _______
 ∫ x^{1}dx = (x^{n+1}/n + 1) + C = x^{0}/0 + C (meaningless)
 natural logarithm

State the Linearity of the Indefinite Integral Theorem
 Sum Rule: ∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx
 Multiples Rule: ∫cf(x)dx = c∫f(x)dx

State the Basic Trigonometric Integrals
 ∫sinx dx = cosx + C
 ∫sec^{2}x dx = tanx + C
 ∫secx tanx dx = secx + C
 ∫cosx dx = sinx + C
 ∫csc^{2}x dx = cotx + C
 ∫cscx cotx dx = cscx + C

Similarly, for any constant k ≠ 0, the formulas:
1) d/dx sin(kx) = k cos(kx)
2) d/dx cos(kx) = ksin(kx)
translate to the which indefinite integral formulas?
 ∫cos(kx)dx = 1/k sin(kx) + C
 ∫sin(kx)dx = 1/k cos(kx) +C

We can think of an antiderivative as a solution to the different equation:
dy/dx = ____
dy/dx = f(x)

Differential equation (Define)
Differential equation: an equation relating an unknown function and its derivatives

The unknown in Eq. (1) is a function y = F(x) whose derivative is _____; that is y = F(x) is an _________ of y = f(x).

Eq. (1) has infinitely many solutions (because the antiderivative is not unique), but we can specify a particular solution by imposing an ______ ______, that is by requiring that the solution satisfy y(x_{0}) = y_{0} for some fixed values x_{0} and y_{0}. A differential equation with an initial condition is called an ______ _____ _______
 initial condition
 initial value problem

s'(t) = _____ and s(t) = ______
v'(t) = _____ and v(t) = ______
 s'(t) = v(t) and s(t) = ∫v(t)dt
 v'(t) = a(t) and v(t) = ∫a(t)dt

