∬Differential Calculus Unit 19 – Antiderivatives

Key Concepts

• Antiderivatives reverse the process of differentiation by finding a function whose derivative is a given function
• The set of all antiderivatives of a function $f(x)$ is denoted by $\int f(x) dx$
• Antiderivatives are not unique; they differ by a constant term $C$ called the constant of integration
• The process of finding antiderivatives is called indefinite integration or antidifferentiation
• Antiderivatives are essential for solving differential equations and calculating definite integrals using the Fundamental Theorem of Calculus
• The power rule, chain rule, and u-substitution are common techniques used to find antiderivatives
• Antiderivatives have numerous applications in physics and engineering, such as determining position from velocity and calculating work done by a variable force

Definition and Notation

• An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$, i.e., $F'(x) = f(x)$
• The indefinite integral notation $\int f(x) dx$ represents the set of all antiderivatives of $f(x)$
  • The symbol $\int$ is called the integral sign, and $f(x)$ is the integrand
  • The variable $x$ is the variable of integration, and $dx$ indicates that the integration is with respect to $x$
• The constant of integration $C$ is added to the antiderivative to represent the entire family of functions that have the same derivative
  • For example, if $F(x) = x^2$, then $\int 2x dx = x^2 + C$, where $C$ can be any real number
• Indefinite integrals can be evaluated using various techniques, such as the power rule, u-substitution, and integration by parts
• The Fundamental Theorem of Calculus connects indefinite integrals (antiderivatives) to definite integrals, which calculate the area under a curve between two points

Properties of Antiderivatives

• Linearity property: $\int [af(x) + bg(x)] dx = a \int f(x) dx + b \int g(x) dx$, where $a$ and $b$ are constants
  • This property allows for the integration of sums and differences of functions
• Constant multiple rule: $\int k f(x) dx = k \int f(x) dx$, where $k$ is a constant
• Power rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$
  • For example, $\int x^3 dx = \frac{x^4}{4} + C$
• Antiderivative of a constant: $\int k dx = kx + C$, where $k$ is a constant
• Antiderivatives are not affected by the addition or subtraction of constants
  • If $F(x)$ is an antiderivative of $f(x)$, then $F(x) + C$ is also an antiderivative of $f(x)$ for any constant $C$
• The antiderivative of a sum or difference of functions is the sum or difference of their antiderivatives
• The chain rule for differentiation becomes the substitution rule for integration

Common Antiderivative Patterns

• Trigonometric functions: $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$
  • Other trigonometric antiderivatives include $\int \sec^2 x dx = \tan x + C$ and $\int \csc^2 x dx = -\cot x + C$
• Exponential functions: $\int e^x dx = e^x + C$, $\int a^x dx = \frac{a^x}{\ln a} + C$ for $a > 0$ and $a \neq 1$
• Logarithmic functions: $\int \frac{1}{x} dx = \ln |x| + C$ for $x \neq 0$
• Inverse trigonometric functions: $\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C$, $\int \frac{1}{1+x^2} dx = \arctan x + C$
• Hyperbolic functions: $\int \sinh x dx = \cosh x + C$, $\int \cosh x dx = \sinh x + C$
• Rational functions: Antiderivatives of rational functions can be found using partial fraction decomposition
  • For example, $\int \frac{2x+1}{x^2+x} dx = \ln |x| + \ln |x+1| + C$

Techniques for Finding Antiderivatives

• u-substitution: A technique for finding antiderivatives of composite functions
  • If $f(x) = g(h(x))h'(x)$, then $\int f(x) dx = \int g(u) du$, where $u = h(x)$
  • This method is the reverse of the chain rule for differentiation
• Integration by parts: A technique for finding antiderivatives of products of functions
  • The formula is $\int u dv = uv - \int v du$
  • This method is useful when the integrand is a product of a function and its derivative
• Partial fraction decomposition: A technique for finding antiderivatives of rational functions
  • The rational function is decomposed into a sum of simpler fractions, which can then be integrated separately
• Trigonometric substitution: A technique for finding antiderivatives of functions involving $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$
  • The substitution $x = a \sin \theta$, $x = a \tan \theta$, or $x = a \sec \theta$ is used, respectively
• Integration by reduction formulas: A technique for finding antiderivatives of functions involving powers of trigonometric functions
  • Reduction formulas are used to express the integral in terms of simpler integrals

Applications in Physics and Engineering

• Position, velocity, and acceleration: Antiderivatives are used to find position from velocity and velocity from acceleration
  • If $v(t)$ is the velocity function, then the position function is $s(t) = \int v(t) dt + C$
  • If $a(t)$ is the acceleration function, then the velocity function is $v(t) = \int a(t) dt + C$
• Work done by a variable force: The work done by a force $F(x)$ along a path from $a$ to $b$ is given by $W = \int_a^b F(x) dx$
• Electric potential and electric field: The electric potential $V(x)$ is the antiderivative of the electric field $E(x)$, i.e., $V(x) = -\int E(x) dx$
• Magnetic vector potential and magnetic field: The magnetic vector potential $\vec{A}$ is related to the magnetic field $\vec{B}$ by $\vec{B} = \nabla \times \vec{A}$
• Fluid dynamics: Antiderivatives are used in the study of fluid flow, such as calculating the velocity potential and stream function
• Heat transfer: Antiderivatives are used to solve heat conduction problems, such as finding the temperature distribution in a material

Relationship to Definite Integrals

• The Fundamental Theorem of Calculus (Part 1) states that if $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$
  • This theorem connects the concept of antiderivatives (indefinite integrals) to definite integrals
• The Fundamental Theorem of Calculus (Part 2) states that if $f$ is continuous on $[a, b]$, then $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
  • This theorem shows that differentiation and integration are inverse processes
• Definite integrals can be evaluated using the Fundamental Theorem of Calculus by finding an antiderivative of the integrand and evaluating it at the limits of integration
• The Net Change Theorem states that $\int_a^b f'(x) dx = f(b) - f(a)$, which is a consequence of the Fundamental Theorem of Calculus
• The Mean Value Theorem for Integrals states that if $f$ is continuous on $[a, b]$, then there exists a point $c$ in $[a, b]$ such that $\int_a^b f(x) dx = f(c)(b - a)$

Practice Problems and Examples

• Find the antiderivative of $f(x) = 3x^2 - 2x + 1$
  • Using the power rule and constant multiple rule, $\int f(x) dx = \int (3x^2 - 2x + 1) dx = x^3 - x^2 + x + C$
• Find the antiderivative of $f(x) = \sin(2x)$
  • Using the chain rule for differentiation in reverse (u-substitution), let $u = 2x$, then $du = 2dx$ or $dx = \frac{1}{2}du$
  • $\int \sin(2x) dx = \int \sin(u) \frac{1}{2}du = -\frac{1}{2}\cos(u) + C = -\frac{1}{2}\cos(2x) + C$
• Evaluate the definite integral $\int_0^1 (x^3 + 2x) dx$
  • First, find the antiderivative: $\int (x^3 + 2x) dx = \frac{x^4}{4} + x^2 + C$
  • Using the Fundamental Theorem of Calculus, evaluate the antiderivative at the limits of integration:
  • $\int_0^1 (x^3 + 2x) dx = [\frac{x^4}{4} + x^2]_0^1 = (\frac{1}{4} + 1) - (0 + 0) = \frac{5}{4}$
• A particle moves along a straight line with velocity $v(t) = t^2 - 4t + 3$. If the particle's initial position is $s(0) = 2$, find its position at time $t$.
  • The position function is the antiderivative of the velocity function: $s(t) = \int v(t) dt + C$
  • $s(t) = \int (t^2 - 4t + 3) dt = \frac{t^3}{3} - 2t^2 + 3t + C$
  • To find the constant $C$, use the initial condition $s(0) = 2$:
    • $2 = s(0) = \frac{0^3}{3} - 2(0)^2 + 3(0) + C$, so $C = 2$
  • The position function is $s(t) = \frac{t^3}{3} - 2t^2 + 3t + 2$