19.1 Definition and basic antiderivatives

Antiderivatives and indefinite integrals are like reverse engineering for functions. Instead of finding how fast something changes, we figure out what function could have led to that rate of change. It's like working backwards from speed to distance.

This process is crucial for solving real-world problems. By understanding antiderivatives, we can predict future values, calculate total changes, and solve complex equations in physics, economics, and engineering.

Antiderivatives and Indefinite Integrals

Antiderivatives and derivatives relationship

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• An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$
  • If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$
  • Example: If $f(x) = 2x$, then $F(x) = x^2 + C$ is an antiderivative of $f(x)$ because $F'(x) = 2x$
• The process of finding an antiderivative is the opposite of finding a derivative
  • Derivatives calculate the rate of change of a function ($f'(x)$ represents the slope of the tangent line at each point)
  • Antiderivatives determine a function given its rate of change ($F(x)$ represents the original function, given the derivative $f(x)$)

Antiderivatives of basic functions

• Power rule: If $f(x) = x^n$, then an antiderivative of $f(x)$ is $F(x) = \frac{x^{n+1}}{n+1} + C$, where $C$ is a constant and $n \neq -1$
  • Example: An antiderivative of $f(x) = x^3$ is $F(x) = \frac{x^4}{4} + C$
  • Example: An antiderivative of $f(x) = \sqrt{x}$ (or $x^{\frac{1}{2}}$) is $F(x) = \frac{2}{3}x^{\frac{3}{2}} + C$
• Exponential rule: If $f(x) = e^x$, then an antiderivative of $f(x)$ is $F(x) = e^x + C$
  • Example: An antiderivative of $f(x) = 3e^x$ is $F(x) = 3e^x + C$
• Trigonometric rules:
  • If $f(x) = \sin(x)$, then an antiderivative of $f(x)$ is $F(x) = -\cos(x) + C$
  • If $f(x) = \cos(x)$, then an antiderivative of $f(x)$ is $F(x) = \sin(x) + C$
  • If $f(x) = \sec^2(x)$, then an antiderivative of $f(x)$ is $F(x) = \tan(x) + C$
  • Example: An antiderivative of $f(x) = 2\sin(x)$ is $F(x) = -2\cos(x) + C$

Rules for complex antiderivatives

• Constant multiple rule: If $F(x)$ is an antiderivative of $f(x)$, then $kF(x)$ is an antiderivative of $kf(x)$, where $k$ is a constant
  • Example: If an antiderivative of $f(x) = x^2$ is $F(x) = \frac{x^3}{3} + C$, then an antiderivative of $3x^2$ is $3(\frac{x^3}{3} + C) = x^3 + C$
  • Example: If an antiderivative of $f(x) = \sin(x)$ is $F(x) = -\cos(x) + C$, then an antiderivative of $5\sin(x)$ is $-5\cos(x) + C$
• Sum rule: If $F(x)$ is an antiderivative of $f(x)$ and $G(x)$ is an antiderivative of $g(x)$, then $F(x) + G(x)$ is an antiderivative of $f(x) + g(x)$
  • Example: If an antiderivative of $f(x) = x^2$ is $F(x) = \frac{x^3}{3} + C$ and an antiderivative of $g(x) = \sin(x)$ is $G(x) = -\cos(x) + C$, then an antiderivative of $x^2 + \sin(x)$ is $\frac{x^3}{3} - \cos(x) + C$
  • Example: An antiderivative of $f(x) = x^3 + e^x$ is $F(x) = \frac{x^4}{4} + e^x + C$

Concept of indefinite integrals

• An indefinite integral is the set of all antiderivatives of a given function
  • The indefinite integral of $f(x)$ is denoted as $\int f(x) \, dx$
  • The variable of integration (usually $x$) is written as $dx$ to indicate the variable with respect to which the integration is performed
  • Example: $\int x^2 \, dx$ represents the set of all antiderivatives of $x^2$
• The result of an indefinite integral includes a constant of integration, typically denoted as $C$
  • Example: $\int x^2 \, dx = \frac{x^3}{3} + C$, where $C$ is an arbitrary constant
  • Example: $\int e^x \, dx = e^x + C$
• The constant of integration represents a family of functions that differ by a constant value
  • The specific value of $C$ is determined by initial conditions or boundary conditions when solving problems involving definite integrals or differential equations
  • Example: If $\int f(x) \, dx = F(x) + C$ and $F(1) = 3$, then $C = 3 - F(1)$