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Step-by-step explanation:
The Reverse of Differentiation
At this point, we know how to find derivatives of various functions. We now ask the opposite question. Given a function f, how can we find a function with derivative f? If we can find a function F derivative f, we call F an antiderivative of f.
Definition: Antiderivative
A function F is an antiderivative of the function f if
F'(x)=f(x)(4.8.1)
for all x in the domain of f.
Consider the function f(x)=2x. Knowing the power rule of differentiation, we conclude that F(x)=x2 is an antiderivative of f since F'(x)=2x. Are there any other antiderivatives of f? Yes; since the derivative of any constant C is zero, x2+C is also an antiderivative of 2x. Therefore, x2+5 and x2−2–√ are also antiderivatives. Are there any others that are not of the form x2+C for some constant C? The answer is no. From Corollary 2 of the Mean Value Theorem, we know that if F and G are differentiable functions such that F'(x)=G'(x), then F(x)−G(x)=C for some constant C. This fact leads to the following important theorem.
General Form of an Antiderivative
Let F be an antiderivative of f over an interval I. Then,
for each constant C, the function F(x)+C is also an antiderivative of f over I;
if G is an antiderivative of f over I, there is a constant C for which G(x)=F(x)+C over I.
In other words, the most general form of the antiderivative of f over I is F(x)+C.
We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.
Example 4.8.1: Finding Antiderivatives
For each of the following functions, find all antiderivatives.
f(x)=3x2
f(x)=1x
f(x)=cosx
f(x)=ex
Solution:
a. Because
ddx(x3)=3x2
then F(x)=x3 is an antiderivative of 3x2. Therefore, every antiderivative of 3x2 is of the form x3+C for some constant C, and every function of the form x3+C is an antiderivative of 3x2.
b. Let f(x)=ln|x|. For x>0,f(x)=ln(x) and
ddx(lnx)=1x.
Forx<0,f(x)=ln(−x) and
ddx(ln(−x))=−1−x=1x.
Therefore,
ddx(ln|x|)=1x.
Thus, F(x)=ln|x| is an antiderivative of 1x. Therefore, every antiderivative of 1x is of the form ln|x|+C for some constant C and every function of the form ln|x|+C is an antiderivative of 1x.
c. We have
ddx(sinx)=cosx,
so F(x)=sinx is an antiderivative of cosx. Therefore, every antiderivative of cosx is of the form sinx+C for some constant C and every function of the form sinx+C is an antiderivative of cosx.
d. Since
ddx(ex)=ex,
then F(x)=ex is an antiderivative of ex. Therefore, every antiderivative of ex is of the form ex+C for some constant C and every function of the form ex+C is an antiderivative of ex.
Exercise 4.8.1
Find all antiderivatives of f(x)=sinx.
Indefinite Integrals
We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function f, we use the notation f'(x) or dfdx to denote the derivative of f. Here we introduce notation for antiderivatives. If F is an antiderivative of f, we say that F(x)+C is the most general antiderivative of f and write
∫f(x)dx=F(x)+C.(4.8.2)
The symbol ∫ is called an integral sign, and ∫f(x)dx is called the indefinite integral of f.
Definition: Indefinite Integrals
Given a function f, the indefinite integral of f, denoted
∫f(x)dx,(4.8.3)
is the most general antiderivative of f. If F is an antiderivative of f, then
∫f(x)dx=F(x)+C.(4.8.4)
The expression f(x) is called the integrand and the variable x is the variable of integration.
Given the terminology introduced in this definition, the act of finding the antiderivatives of a function f is usually referred to as integrating f.
For a function f and an antiderivative F, the functions F(x)+C, where C is any real number, is often referred to as the family of antiderivatives of f. For example, since x2 is an antiderivative of 2x and any antiderivative of 2x is of the form x2+C, we write
∫2xdx=x2+C.(4.8.5)
The collection of all functions of the form x2+C, where C is any real number, is known as the family of antiderivatives of 2x. Figure shows a graph of this family of antiderivatives.
Figure 4.8.1: The family of antiderivatives of 2x consists of all functions of the form x2+C, where C is any real number.
For some functions, evaluating indefinite integrals follows directly from properties of derivatives. For example, for n≠−1,
∫xndx=xn+1n+1+C,
which comes directly from
ddx(xn+1n+1)=(n+1)xnn+1=xn.
This fact is known as the power rule for integrals.
Power Rule for Integrals
For n≠−1,
∫xndx=xn+1n+1+C.(4.8.6)
Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in Appendix B.
