Lecture #1: Antiderivatives

Antiderivatives

• Scope of Calculus 2: Calculus 1 focuses on differential calculus (derivatives). Calculus 2 focuses on integral calculus: understanding the integral, developing analytical techniques to solve integrals, and utilizing integrals to resolve real-world physical problems.

• Kinematic Motivation of the Antiderivative:

  • Position Function: $s(t)$ describes the spatial location of an object at time $t$. For instance, $s(0) = 0$ indicates an initial position at the origin, whereas $s(2) = 3$ indicates the object has displaced $3\text{ units}$ at $t = 2$.

  • Velocity as a Derivative: Velocity $v(t)$ represents the rate of change of position with respect to time: $s'(t) = v(t)$.

  • Acceleration as a Derivative: Acceleration $a(t)$ represents the rate of change of velocity with respect to time: $v'(t) = a(t)$.

  • Inversion of Kinematics: Determining position from velocity requires inverting differentiation. Position $s(t)$ is an antiderivative of velocity $v(t)$. Likewise, velocity $v(t)$ is an antiderivative of acceleration $a(t)$.

• Formal Definition of an Antiderivative: A function $F(x)$ is an antiderivative of a function $f(x)$ on an interval $I$ if and only if:   $F'(x) = f(x)$   for all $x$ within $I$.

• Multiplicity of Antiderivatives and the Arbitrary Constant:

  • Differentiation maps constants to zero: $\frac{d}{dx}[C] = 0$.

  • For the constant function $f(x) = 3$, determining the antiderivative requires solving for a function whose derivative is $3$.

  • Differentiating $F(x) = 3x$ yields $F'(x) = 3$.

  • Functions such as $F(x) = 3x + 1$, $F(x) = 3x + 5$, $F(x) = 3x + 100$, or $F(x) = 3x + 1000000$ also differentiate to $3$.

  • Consequently, any specific function has infinitely many antiderivatives differing only by an additive constant.

• General Form Theorem: If $F(x)$ is an antiderivative of $f(x)$ over an interval $I$, then for any constant $C$, the general antiderivative is:   $F(x) + C$

• Constant Difference Theorem: If two functions $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$ over an interval $I$, then their difference is identically constant:   $G(x) - F(x) = C \implies G(x) = F(x) + C$

The Indefinite Integral and Operational Rules

• Definition of the Indefinite Integral: The collection of all antiderivatives of a function $f(x)$ is denoted using the indefinite integral symbol:   $\int f(x)\,dx = F(x) + C$

  • Integral Sign ($\int$): An elongated "S" symbolizing summation.

  • Integrand ($f(x)$): The function undergoing integration.

  • Differential ($dx$): Identifies the variable of integration (the active variable of interest).

  • Constant of Integration ($C$): An arbitrary, undecided real constant.

  • Indefinite integrals omit upper and lower limits of integration, yielding a family of functions, whereas definite integrals evaluate across specific boundary limits.

• Power Rule for Integration: Derived by reversing the power rule of differentiation ($\frac{d}{dx}[x^{n+1}] = (n+1)x^n$):   $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (\text{for } n \neq -1)$   The factor $n+1$ in the denominator counteracts the multiplication generated when differentiating the raised power.

• Linearity Properties of Indefinite Integrals:

  • Sum and Difference Rule: The integral of a finite sum or difference equals the sum or difference of the individual integrals:     $\int (f(x) \pm g(x))\,dx = \int f(x)\,dx \pm \int g(x)\,dx$

  • Constant Multiple Rule: Constant scalar factors can be factored out of the integral:     $\int c \cdot f(x)\,dx = c \int f(x)\,dx$

  • When computing integrals composed of multiple terms, separate integration constants combine into a single, global arbitrary constant $C$.

Standard Integration Formulas and Worked Examples

• Essential Formulas for Examination Mastery (OpenStax Volume 1, Table 4.3 / 4.13; formulas are not provided on exams):

  • $\int \cos(x)\,dx = \sin(x) + C$

  • $\int \sin(x)\,dx = -\cos(x) + C$

  • $\int \sec^2(x)\,dx = \tan(x) + C$

  • $\int \frac{1}{\sqrt{1 - x^2}}\,dx = \arcsin(x) + C$

  • $\int \frac{1}{1 + x^2}\,dx = \arctan(x) + C$

• Example 1 (Monomial Antiderivative):

  • Problem: Evaluate $\int 3x^2\,dx$.

  • Procedure: Apply the inverted power rule by advancing the exponent from $2$ to $3$. Differentiating $x^3$ produces $3x^2$, which matches the integrand.

  • Solution:     $\int 3x^2\,dx = x^3 + C$

• Example 2 (Reciprocal Function and the Domain Matching Principle):

  • Problem: Evaluate $\int \frac{1}{x}\,dx$.

  • Theoretical Foundation: Differentiating $\ln(x)$ yields $\frac{1}{x}$.

  • Domain Discrepancy Trap: The domain of $f(x) = \frac{1}{x}$ is all non-zero real numbers ($x \neq 0$), whereas the standard natural logarithm $\ln(x)$ is defined strictly for $x > 0$.

  • Resolution: To reconcile the mathematical domains, the absolute value must enclose $x$:     $\int \frac{1}{x}\,dx = \ln(|x|) + C$

• Example 3 (Basic Trigonometric Antiderivative):

  • Problem: Evaluate $\int \cos(x)\,dx$.

  • Procedure: Identify the function whose derivative yields $\cos(x)$. Since $\frac{d}{dx}[\sin(x)] = \cos(x)$, the evaluation holds directly.

  • Solution:     $\int \cos(x)\,dx = \sin(x) + C$

• Example 4 (Linear Combination of Polynomial and Trigonometric Terms):

  • Problem: Evaluate $\int (3x^2 + \cos(x))\,dx$.

  • Procedure: Separate the integrand using the sum rule:     $\int (3x^2 + \cos(x))\,dx = \int 3x^2\,dx + \int \cos(x)\,dx = x^3 + \sin(x) + C$

  • Derivative Verification: Let $F(x) = x^3 + \sin(x) + C$. Differentiating gives $F'(x) = 3x^2 + \cos(x) + 0$, confirming the original integrand.

• Example 5 (Exponential Functions and Chain Rule Balancing):

  • Standard Base Case:     $\int e^x\,dx = e^x + C$

  • Composite Argument Case: Evaluate $\int e^{2x}\,dx$.

  • Analysis via Chain Rule: Testing $F(x) = e^{2x}$ yields a derivative of $F'(x) = 2e^{2x}$, which is twice the target integrand.

  • Scalar Adjustment: Scale the initial guess by $\frac{1}{2}$:     $\frac{d}{dx}\left[\frac{1}{2}e^{2x}\right] = \frac{1}{2} \cdot 2e^{2x} = e^{2x}$

  • Solution:     $\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C$

• Example 6 (Multi-Term Polynomial Antidifferentiation):

  • Problem: Evaluate $\int (4x^3 + 5x^2 - x - 7)\,dx$.

  • Component-by-Component Evaluation:

  • Term 1: $\int 4x^3\,dx = 4\left(\frac{x^4}{4}\right) = x^4$

  • Term 2: $\int 5x^2\,dx = 5\left(\frac{x^3}{3}\right) = \frac{5}{3}x^3$

  • Term 3: $\int -x\,dx = -\frac{x^2}{2} = -\frac{1}{2}x^2$

  • Term 4: $\int -7\,dx = -7x$

  • Complete Antiderivative:     $\int (4x^3 + 5x^2 - x - 7)\,dx = x^4 + \frac{5}{3}x^3 - \frac{1}{2}x^2 - 7x + C$

  • Cautionary Note: Omitting $+ C$ is a frequent error on the Basic Skills Exam leading directly to lost points.

• Example 7 (Algebraic Decomposition via Rational Exponents):

  • Problem Structure: Rational expressions with multi-term numerators and monomial denominators, such as $\int \frac{x^2 + 4x^{1/3}}{x}\,dx$.

  • Algebraic Reduction Technique: Divide each numerator term by the denominator before integrating:     $\frac{x^2 + 4x^{1/3}}{x} = \frac{x^2}{x} + \frac{4x^{1/3}}{x} = x + 4x^{\frac{1}{3} - 1} = x + 4x^{-\frac{2}{3}}$

  • Subsequent Integration: Apply the standard power rule to each simplified term individually:     $\int \left(x + 4x^{-\frac{2}{3}}\right)\,dx = \frac{x^2}{2} + 4\left(\frac{x^{-\frac{2}{3} + 1}}{-\frac{2}{3} + 1}\right) + C = \frac{1}{2}x^2 + 12x^{\frac{1}{3}} + C$