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The Integral

1 Antiderivatives and the Indefinite Integral

Suppose that we are given the function $f(x)=3$ and we are asked to find its antiderivative. We must find a function $F(x)$ whose derivative is $f(x).$It is easy to see that one such function is$F(x)=3x.$Notice that antiderivatives are not unique, however, since $G(x)=3x+5$ is also an antiderivative of $f(x)$ as is $H(x)=3x-7.$In fact, any function $F(x)=3x+C$, where C is a real number constant, is an antiderivative of $f(x).$Another way of saying this is to say that any function whose derivative is given by the constant $3$ must have the form $3x+C.$ In general, any two functions with the same derivative will differ only by a constant. That is, if $F(x)$ is an antiderivative of $f(x),$then $F(x)+C$ is also an antiderivative for any constant $C.$

Definition: A function $F(x)$ is called an antiderivative of $f(x)$ if MATH




Notation: If MATHthen $F(x)+C$ is called the general antiderivative of $f(x).$ The process of finding antiderivatives is called integration.We write $\int f(x)dx=F(x)+C$ to denote the general antiderivative of a function $f(x).$ $\int f(x)dx$ is called the indefinite integral of the function $f$ with respect to $x$ and $\int $is called an integral sign. The function $f(x)$ is called the integrand and $C$ is called the constant of integration. In summary, MATH


Example 1

Find the general antiderative of $f(x)=x^{2}.$

Solution: We need to find a function whose derivative is $x^{2}.$ Since $3x^{2}$ is the derivative of $x^{3},$ we see that an antiderivative of $x^{2}$ is MATHAll other antiderivatives of $f(x)$ will have the formMATH


We can check this answer by taking the derivative of $F(x):$MATH

Example 2

Evaluate $\int -7dx.$

Solution: We are being asked to find the indefinite integral of the function $f(x)=-7$ with respect to $x.$That is, we must find an antiderivative of $-7.$Since

the derivative of $-7x$ is $-7,$we see that MATH

Again, we can check the answer by differentiating:MATH





2 Basic Integration Formulas

Since integration or antidifferentiation is the reverse process of differentiation, we can use the derivative rules in reverse to find indefinite integrals. Not all of the differentiation formulas that we learned in Lesson 6 are easy to state in reverse. The following list gives the basic integration formulas.

1. The general antiderivative of a constant $k$ is $kx+C.\ \ \ \ \ $
MATH
2. The general antiderivative of a power function $x^{n}$ is MATH provided $n\neq -1.$
MATH (Add 1 to the power and divide by the new power.)
3. An antiderivative of a sum (or difference) of terms is the sum (or difference)of the antiderivatives.
MATH
4. An antiderivative for a constant times a function is the constant times the antiderivative of the function.
MATH
5. The general antiderivative of $\frac{1}{x}$ is $\ln x.$
MATH
6. The general antiderivative for $e^{x}$ is $e^{x}.$
MATH
7. The general antiderivative for $b^{x}$ is MATH
MATH




The following examples illustrate how the Integration formulas work:


Example 1 Evaluate $\int x^{5}dx$

Solution Using formula 2 above, we get MATH


Since integration is the reverse process of differentiation, we can check this answer by taking the derivative of our answer.MATH

Since $x^{5}$
is the integrand in our original problem, we know that we have found the correct answer.


Example 2 Evaluate $\int 8x^{3}dx$

Solution MATH


Again, we can check this answer by differentiationMATH


Example 3 Evaluate MATH

Solution MATH

Check by differentiating:MATH


Example 4 Evaluate MATH

Solution MATHcheck by differentiating:MATH


Example 5 Evaluate MATH

Solution MATH


Check the answer by differentiating:MATH


Example 6 Evaluate MATH

Solution MATH

Check by differentiating:MATH



3 The Definite Integral

The definite integral from a to b of of a continuous function $f(x)$ with respect to $x$ is MATHwhere MATHNotice that we use the same notation as for the indefinte integral except that we add the numbers $a$ and $b$ at the bottom and top of the integral sign. The numbers $a$ and $b$ are called the limits of integration, $f(x)$ is the integrand, and $dx$ is simply notation for "with respect to x". So, $\int_{a}^{b}f(x)dx$ is read as "the definite integral of $f(x)$ with respect to $x'.$



Example 7

Evaluate MATH.

Solution

First we need to find an antiderivative for $f(x)=3x^{2}-2x+3.$ The antiderivative is MATH (we will choose to let $C=0).$ ThenMATH




Example 8

Evaluate MATH.

Solution

An antiderivative for $f(x)=x^{3}-3x-1$ is MATH So,

MATH