3.10: Antiderivatives

3.10 Antiderivatives

Learning Objectives

• Find the general antiderivative of a given function by applying integration techniques, including basic rules for integrals.
• Apply basic rules for integrals (e.g., power rule, sum rule) to compute antiderivatives of functions.
• Solve initial-value problems by finding the antiderivative of a function and applying the given initial conditions to determine the constant of integration.

Definition \(\PageIndex{9}\)

A function \( F \) is called an antiderivative of \( f \) on an interval \( I \) if \( F'(x) = f(x) \ \text{for all } x \text{ in } I.\)

For example, if \( f(x) = x^2 \), then \( F(x) = \frac{x^3}{3} \) is an antiderivative of \( f \).

Theorem \(\PageIndex{9}\)

Suppose that \( F \) is an antiderivative of \( f \) on an interval \( I \). Then the most general antiderivative of \( f \) on \( I \) is:

\[
F(x) + C
\]

where \( C \) is a constant. Furthermore, if \( G \) is an antiderivative of \( f \) over \( I \), then there exists a constant \( C \) such that:

\[
G(x) = F(x) + C \quad \text{on } I
\]

Example \(\PageIndex{32}\)

Find the general antiderivative of the following functions:

1. \( f(x) = e^x \)
2. \( f(x) = \sin x \)
3. \( f(x) = \cos x \)
4. \( f(x) = \dfrac{1}{x} \)

Definition \(\PageIndex{10}\)

Let \( f \) be a function. The indefinite integral of \( f \), denoted

\[
\int f(x) \, dx
\]

is defined as the most general antiderivative of \( f \). That is, if \( F \) is an antiderivative of \( f \), then

\[
\int f(x) \, dx = F(x) + C
\]

where \( C \) is an arbitrary constant.

Derivative Formula	Indefinite Integral
\( \dfrac{d}{dx}[k] = 0 \)	\( \displaystyle \int k \, dx = kx + C \)
\( \dfrac{d}{dx}[x^n] = nx^{n-1} \)	\( \displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), for \( n \ne -1 \)
\( \dfrac{d}{dx}[\ln |x|] = \dfrac{1}{x} \)	\( \displaystyle \int \frac{1}{x} \, dx = \ln |x| + C \)
\( \dfrac{d}{dx}[e^x] = e^x \)	\( \displaystyle \int e^x \, dx = e^x + C \)
\( \dfrac{d}{dx}[\sin x] = \cos x \)	\( \displaystyle \int \cos x \, dx = \sin x + C \)
\( \dfrac{d}{dx}[\cos x] = -\sin x \)	\( \displaystyle \int \sin x \, dx = -\cos x + C \)
\( \dfrac{d}{dx}[\tan x] = \sec^2 x \)	\( \displaystyle \int \sec^2 x \, dx = \tan x + C \)
\( \dfrac{d}{dx}[\sin^{-1} x] = \dfrac{1}{\sqrt{1 - x^2}}\)	\( \displaystyle \int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1} x + C \)
\( \dfrac{d}{dx}[\cos^{-1} x] = -\dfrac{1}{\sqrt{1 - x^2}} \)	\( \displaystyle \int -\frac{1}{\sqrt{1 - x^2}} \, dx = \cos^{-1} x + C \)
\( \dfrac{d}{dx}[\tan^{-1} x] = \dfrac{1}{1 + x^2} \)	\( \displaystyle \int \frac{1}{1 + x^2} \, dx = \tan^{-1} x + C \)

Theorem \(\PageIndex{10}\)

Properties of Indefinite Integrals:

Let \( F \) and \( G \) be antiderivatives of \( f \) and \( g \), respectively. Let \( k \) be a constant.

1. \( \displaystyle \int (f(x) \pm g(x)) \, dx = F(x) \pm G(x) + C \)
2. \( \displaystyle \int k f(x) \, dx = k F(x) + C \)

Example \(\PageIndex{33}\)

Evaluate the following indefinite integrals:

1. \( \displaystyle \int (8x^9 + 2x^6 - 4x^2 - 5) \, dx \)
2. \( \displaystyle \int (4x^5 + e^x) \, dx \)
3. \( \displaystyle \int \frac{3x^5 + 10x^3 + 4x}{x^5} \, dx \)
4. \( \displaystyle \int \frac{3x^5 + 7\sqrt[5]{x} + 2}{x} \, dx \)
5. \( \displaystyle \int \frac{2}{\sqrt{1 - x^2}} \, dx \)
6. \( \displaystyle \int (\cos x - 4\sec^2 x) \, dx \)

Initial-value problem

A differential equation is an equation that relates an unknown function and one or more of its derivatives. The equation

\[\dfrac{dy}{dx}=f(x)\label{diffeq1} \]

is a simple example of a differential equation. Solving this equation means finding a function \(y\) with a derivative \(f\). Therefore, the solutions of Equation \ref{diffeq1} are the antiderivatives of \(f\). If \(F\) is one antiderivative of \( f\), every function of the form \( y=F(x)+C\) is a solution of that differential equation. For example, the solutions of

\[\dfrac{dy}{dx}=4x^3\nonumber \]

are given by

\[y=\int 4x^3\,dx=x^4+C.\nonumber \]

Sometimes we are interested in determining whether a particular solution curve passes through a certain point \( (x_0,y_0)\) —that is, \( y(x_0)=y_0\). The problem of finding a function \(y\) that satisfies a differential equation

\(\dfrac{dy}{dx}=f(x)\)

with the additional condition

\(y(x_0)=y_0\)

is an example of an initial-value problem. The condition \( y(x_0)=y_0\) is known as an initial condition. For example, looking for a function \( y\) that satisfies the differential equation

\(\dfrac{dy}{dx}=4x^3\)

and the initial condition

\(y(2)=6\)

is an example of an initial-value problem. Since the solutions of the differential equation are \( y=x^4+C,\) to find a function \(y\) that also satisfies the initial condition, we need to find \(C\) such that \(y(2)=(2)^4+C=6\). From this equation, we see that \( C=-10\), and we conclude that \( y=x^4-10\) is the solution of this initial-value problem.

Example \(\PageIndex{34}\)

Solve the following initial value problems:

1. \( \dfrac{dy}{dx} = 4 + \sqrt{x}, \quad y(4) = 20 \)
2. \( \dfrac{dy}{dx} = 4x^5 + e^x, \quad y(1) = e \)
3. \( \dfrac{dy}{dx} = \cos x - 4\sec^2 x, \quad y\left(\dfrac{\pi}{4}\right) = \dfrac{\sqrt{2}}{2} \)

Example \(\PageIndex{35}\)

A particle in motion has acceleration given by:

\[
a(t) = 12t^2 - 6 \ \text{and} \ s(2) = 5.
\]

Find the position function \( s(t) \) of the particle.

Example \(\PageIndex{36}\)

A car is traveling at a speed of 66 ft/sec (45 mph) when the brakes are applied. The car begins decelerating at a constant rate of 10 ft/sec².

1. How long does it take for the car to stop?
2. What distance does the car cover before stopping?