4.1 Antiderivatives and Indefinite Integrals

Defn. A function F(x) is an antiderivative of f(x) on an interval I if F '(x)=f(x) for all x in I.

ex. Find an antiderivative of f(x)=x2

ex. Find an antiderivative of f(x)=sinx

ex. Find an antiderivative of f(x)=

Thm. If F is an antiderivative of f on an interval I, then G is an antiderivative of f iff G is of the form G(x)=F(x)+c, where c is a constant.

G is called a General Antiderivative.c is called the Constant of Integration.

Defn. A Differential Equation in x and y is an equation that involves derivatives of y.

These all mean the same thing:

Find the general solution of a differential equation.

Find an antiderivative of a function.

Evaluate an indefinite integral.

So… how many antiderivatives of f are there?

An infinite number

ex. Find a general solution for the differential equation y'=3x.

ex. Find the particular solution for the differential equation

y'=3x if x=1 when y=2/3.

Notation

= f '(x)

dy= f '(x) dx

This is an antiderivative symbol, also called an integral symbol. It's an elongated S.

}

integrand

This shows the variable of integration.

}

The antiderivative

The constant of integration.

Sometimes the integrand needs to be rewritten to be more useful.

Defn. The operation of finding antiderivatives is called indefinite integration.

Basic Integration Rules (p.250)

So, integration and differentiation are inverse operations (just like + and -, x and ÷, or squaring and square rooting).

For some constant k,

n

If dy/dx = 2x-1 what is the general solution of this differential equation?

What is the particular solution of this differential equation for x=1 and y=1?

Suppose that the acceleration of a particle is a(t)= Find its position function x(t) if v(4)=2 and x(0)=0.

HW: p256#3-42x3