Section 10.1 Antiderivatives
¶Subsection 10.1.1 Terminology
Definition 10.1.1. Antiderivatives.
Given a function f(x)\text{,} we say that F(x) is an antiderivative of f(x) if f(x) is the derivative of F(x)\text{.} That is, F'(x)=f(x)\text{.}
Example 10.1.2.
Compare the following derivatives:
Each of the functions have the same derivative. We say that x^2+3x\text{,} x^2+3x-1\text{,} and x^2+3x+4 are all antiderivatives of 2x+3\text{.} More generally, we know x^2+3x+C will be an antiderivative for any constant value C\text{.}
Theorem 10.1.3.
Suppose that F(x) and G(x) are both antiderivatives of f(x) on an interval I\text{.} That is, for all x \in I we have
Then there is a constant C so that for all x \in I\text{,} G(x) = F(x)+C\text{.}
Definition 10.1.4. Indefinite Integrals.
Given a function f(x)\text{,} the indefinite integral of f(x) with respect to x\text{,} written \displaystyle \int f(x) \, dx\text{,} is the general antiderivative of f(x)\text{.} That is, if F(x) is any antiderivative such that F'(x)=f(x)\text{,} then
Subsection 10.1.2 Examples
For the most part, finding antiderivatives corresponds to recognizing how a function might have been computed as a derivative. Every statement about differentiation has an equivalent statement about integrals. To check whether a proposed function is an antiderivative, we calculate its derivative and compare that with the function inside the integral.Example 10.1.5.
Find \displaystyle \frac{d}{dx}\Big[ (3x+5)^4 \Big] and then write down the equivalent statement as an integral.
The last operation in the expression \((3x+5)^4\) is the power acting on the expression \(u=3x+5\text{.}\) The derivative requires a chain rule:
Once we know the derivative, we can write the equivalent integral
This says that \((3x+5)^4\) is an antiderivative of \(12(3x+5)^3\text{,}\) along with that same formula plus any constant.
Theorem 10.1.6.
If F(x) is an antiderivative of f(x) and G(x) is an antiderivative of g(x)\text{,} then for any constants c_1 and c_2\text{,} c_1 F(x) + c_2 G(x) is an antiderivative of c_1 f(x) + c_2 g(x)\text{.} We write
Example 10.1.7.
Find \displaystyle \int 4x^3 - 2e^{2x} \, dx\text{.}
We are looking for a function \(F(x)\) for which \(F'(x)=4x^3 - 2e^{2x}\text{.}\) From experience computing derivatives, we know
This suggests we should use the difference \(F(x) = x^4-e^{2x}\text{.}\) We verify by differentiation:
This verifies that \(F(x)\) is an antiderivative of \(4x^3-2e^{2x}\text{.}\) The general antiderivative is written as the indefinite integral,
Example 10.1.8.
Find \displaystyle \int x^2(x^2-3) \, dx\text{.}
The function \(f(x) = x^2(x^2-3)\) is a product that can be expanded to a sum using the distributive property.
Our experience with the power rule suggests that we should be able to integrate this expression. We know
To eliminate the unwanted constant multiple of 5, we can multiply both sides by \(\frac{1}{5}\) to get
This suggests an antiderivative
We verify using regular differentiation rules:
We have found
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Power Rule: For any power n \ne -1\text{,}
\begin{equation*} \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C. \end{equation*} -
Generalized Power Rule: For any power n \ne -1 and expression u\text{,}
\begin{equation*} \int u^n \cdot \frac{du}{dx} \, dx = \frac{1}{n+1} u^{n+1} + C. \end{equation*} -
Logarithm Rule:
\begin{equation*} \int \frac{1}{x} \, dx = \ln(|x|) + C. \end{equation*} -
Generalized Logarithm Rule: For any expression u\text{,}
\begin{equation*} \int \frac{u'}{u} \, dx = \ln(|u|) + C. \end{equation*} -
Elementary Exponential Rule: For any real value k \ne 0\text{,}
\begin{equation*} \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C. \end{equation*} -
Generalized Exponential Rule: For any expression u\text{,}
\begin{equation*} \int e^{u} \cdot \frac{du}{dx} \, dx = e^{u} + C. \end{equation*}
Example 10.1.9.
\displaystyle \int x^2 e^{x^3} \, dx
Because the integrand has a product of expressions, we should begin by looking to see if the problem involves the chain rule. The exponential term \(e^{x^3}\) involves the expression \(u=x^3\) which has a derivative \(u'=3x^2\text{.}\) Notice that the other factor in the problem, \(x^2\text{,}\) differs from \(u'\) only by a constant multiple. That is, we can recognize our problem as a generalized exponential
Subsection 10.1.3 Finding a Particular Antiderivative
Adding a constant to a function represents a graphical transformation of a vertical shift. Consequently, different antiderivatives have the same graph shifted vertically from one another. Consider the function f(x) = x^2 - 4x\text{.} Integration gives usExample 10.1.11.
Find the constant C so that F(x) = \frac{1}{3}x^3 - 2x^2 + C satisfies F(3)=2\text{.}
Substitute the value \(x=3\) into the equation for \(F(x)\text{.}\)
Because we want \(F(3)=2\text{,}\) we create the equation
so that we can solve for \(C\) to get \(C=11\text{.}\)
Example 10.1.12.
Find a function P(t) so that P'(t) = 20e^{-2t} + 3t and P(0) = 50\text{.}
Start by finding the general antiderivative.
We therefore see that \(P(t) = -10 e^{-2t} + \frac{3}{2}t^2 + C\text{.}\) Now we substitute \(t=0\) and \(P(0)=50\) to solve for \(C\text{.}\)
Having found \(C=60\text{,}\) we can conclude
Example 10.1.13.
A cup of coffee starts at a temperature of 160 degrees Fahrenheit. The temperature changes at a rate of change (degrees per minute) modeled by the formula -3.6e^{-0.04t} where t is the time in minutes. Find the temperature as a function of time.
Let \(T\) represent the temperature of the cup of coffee in degrees Fahrenheit. Our given information shows that
The temperature \(T\) must be an antiderivative of this formula,
To find the value of \(C\text{,}\) substitute \(t=0\) and \(T=160\text{.}\)
Consequently, we have \(T = 90e^{-0.04t} + 70\text{.}\)
Subsection 10.1.4 Summary
An antiderivative of f(x) is any function F(x) so that \frac{d}{dx}[F(x)] = f(x)\text{.} If F(x) is an antiderivative of f(x)\text{,} then so is F(x)+C for any value of C\text{.}
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The Fundamental Theorem of Calculus guarantees that every continuous function has an antiderivative. In particular, if f(x) is continuous on an interval I with a \in I\text{,} then the accumulation function
\begin{equation*} A(x) = \int_a^x f(z) \, dz \end{equation*}is an antiderivative on the interval I\text{.}
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We use the indefinite integral as the operator for antidifferentiation. For a function f(x) with antiderivative F(x)\text{,} we write
\begin{equation*} \int [ f(x) ]\, dx = F(x) + C \end{equation*}where C (or any other chosen symbol) represents an arbitrary constant of integration.
The constant of integration graphically represents an arbitrary vertical shift of the graph of a function. Given any point representing an initial value, we can solve for the constant of integration so that there is the graph of an antiderivative which passes through the given point.
Exercises 10.1.5 Exercises
1.
\displaystyle \frac{d}{dx}\Big[ 2x^4 \Big]
2.
\displaystyle \frac{d}{dx}\Big[ (2x+3)^5 \Big]
3.
\displaystyle \frac{d}{dx}\Big[ \sqrt{x^2+3} \Big]
4.
\displaystyle \frac{d}{dx}\Big[ \ln(|x^2-4x|) \Big]
5.
\displaystyle \frac{d}{dx}\Big[ e^{3x^4} \Big]
6.
\displaystyle \frac{d}{dx}\Big[ x^2 e^{-3x} \Big]
7.
\displaystyle \frac{d}{dx}\Big[ x \ln(|x|) \Big]
8.
\displaystyle \frac{d}{dx}\Big[ \frac{x-1}{x-3} \Big]
9.
\displaystyle \int -3x^5 + 2x^2 + 3 \,dx
10.
\displaystyle \int 2x - 4x^{-1} + 5x^{-3} \,dx
11.
\displaystyle \int x^3(3x^2-4x+7) \,dx
12.
\displaystyle \int (x+4)(x-8) \,dx
13.
\displaystyle \int \frac{x^2+4x-5}{3x^2} \,dx
14.
\displaystyle \int e^{2x} \,dx
15.
\displaystyle \int 4e^{-3x} \,dx
16.
\displaystyle \int xe^{x^2} \,dx
17.
\displaystyle \int 2x^3e^{-x^4} \,dx
18.
\displaystyle \int \frac{1}{x+3} \,dx
19.
\displaystyle \int \frac{3}{2x+1} \,dx
20.
\displaystyle \int \frac{x}{x^2+4} \,dx
21.
\displaystyle \int \frac{e^{2x}}{e^{2x}+1} \,dx
22.
\displaystyle \int -xe^{-x} + e^{-x} \,dx
23.
\displaystyle \int \frac{2xe^{2x}-e^{2x}}{x^2} \,dx
24.
Find f(x) if f'(x) = 2x-5 with f(1)=4\text{.}
25.
Find g(x) if g'(x) = 3e^{-3x} with g(0)=2\text{.}
26.
The velocity of a vehicle on track that runs left to right is v(t) = \frac{1}{2}t^2 - 8t + 24\text{.} If the vehicle is at a position s=0 when t=1\text{,} find the position s(t) as a function of time.
27.
A population changes at a rate defined by R(t) = 0.24t^2 - 24 t + 216\text{,} where t is measured in years. If the population is P=120000 when t=0\text{,} find the population as a function of time.
28.
A radiation detector absorbs radiation at a rate of R(t) = 5e^{-0.1t} (grays per minute). Find the total amount of radiation absorbed by the detector as a function of time t (minutes) since t=0\text{.}