THE INDEFINITE INTEGRAL AND DIFFERENTIAL EQUATIONS

THE INDEFINITE INTEGRAL AND ANTI-DERIVATIVES

• Definition 1.1: Antiderivative (Indefinite Integral)     * A function $F$ is called an antiderivative (also known as indefinite integral) of the function $f$ on an interval $I$ if $F'(x) = f(x)$ for all $x \in I$.

• Theorem 1.1: General Form of Antiderivatives     * If $F$ is a particular antiderivative of $f$ on an interval $I$, then every antiderivative of $F$ is of the form $F(x) + C$, where $C$ is an arbitrary constant.     * Any particular antiderivative can be obtained by assigning a particular value for $C$.

• Definition 1.2: Antidifferentiation (Integration)     * Antidifferentiation is an operation that finds all antiderivatives of a function and is denoted by the integral sign $\int$.     * Suppose $F'(x) = f(x)$ and $d(F(x)) = f(x)dx$, then $\int f(x)dx = F(x) + C$ for any constant $C$.     * The Constant of Integration: The constant $C$ is formally called the constant of integration.     * Nomenclature: The expression $\int f(x)dx$ is read verbally as “integral of $f(x)dx$”.

BASIC INTEGRATION FORMULAS AND THE POWER RULE

• Theorem 1.2: The Power Rule For Integration     * Let $n$ be a rational number such that $n \neq 1$.     * The integral is defined as: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for some constant $C$.

• Proof of Theorem 1.2:     * Let $F(x) = \frac{x^{n+1}}{n+1} + C$, for some constant $C$.     * Then $F'(x) = \frac{d}{dx} (\frac{x^{n+1}}{n+1} + C)$.     * Applying linearity and derivative rules: $F'(x) = \frac{1}{n+1} \times \frac{d}{dx} (x^{n+1}) + \frac{d}{dx} (C)$.     * $F'(x) = \frac{1}{n+1} \times (n+1)x^{(n+1)-1} = x^n$.     * Therefore, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.

ADDITIONAL INTEGRATION FORMULAS

• Basic and Algebraic Formulas:     * $\int dx = x + C$     * $\int e^x dx = e^x + C$     * $\int \frac{dx}{x} = \ln|x| + C$, for $x \neq 0$     * $\int a^x dx = a^x \ln a + C$, for $a > 0$ and $a \neq 1$

• Trigonometric Formulas:     * $\int \cos(x) dx = \sin(x) + C$     * $\int \sin(x) dx = -\cos(x) + C$     * $\int \sec^2(x) dx = \tan(x) + C$     * $\int \csc^2(x) dx = -\cot(x) + C$     * $\int \sec(x) \tan(x) dx = \sec(x) + C$     * $\int \csc(x) \cot(x) dx = -\csc(x) + C$

• Logarithmic Trigonometric Integrals:     * $\int \tan(x) dx = \ln|\sec(x)| + C$     * $\int \cot(x) dx = \ln|\sin(x)| + C$     * $\int \sec(x) dx = \ln|\sec(x) + \tan(x)| + C$     * $\int \csc(x) dx = \ln|\csc(x) - \cot(x)| + C$

PROPERTIES OF THE INDEFINITE INTEGRAL

• Theorem 1.3: Linearity of the Integral     * Let $f$ and $g$ be functions.     * Constant Multiple Rule: $\int cf(x)dx = c \int f(x) dx$, for any constant $c$.     * Sum Rule: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$.

• Proof of Theorem 1.3:     * Let $F$ and $G$ be antiderivatives of $f$ and $g$, respectively ($F'(x) = f(x)$ and $G'(x) = g(x)$).     * Equivalently, $\int f(x) dx = F(x) + C_1$ and $\int g(x) dx = G(x) + C_2$.     * For the constant multiple rule: $\frac{d}{dx} (c \int f(x) dx) = c \frac{d}{dx} (F(x) + C_1) = cF'(x) = cf(x)$. Thus, $\int cf(x)dx = c \int f(x) dx$.     * For the sum rule: $\frac{d}{dx} (\int f(x) dx + \int g(x) dx) = \frac{d}{dx} (F(x) + G(x) + C_1 + C_2) = F'(x) + G'(x) = f(x) + g(x)$. Thus, $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$.

• Generalized Extension:     * If $f_1, f_2, \dots, f_n$ are functions on the same interval, then for any constants $c_1, c_2, \dots, c_n$:     * $\int [c_1f_1(x) + c_2f_2(x) + \dots + c_nf_n(x)] dx = c_1 \int f_1(x)dx + c_2 \int f_2(x)dx + \dots + c_n \int f_n(x)dx$.

THE INTEGRAL OF A COMPOSITE FUNCTION AND THE CHAIN RULE

• Theorem 1.4: Composite Function Rule     * Let $g$ be a differentiable function with range as an interval $I$. Suppose $f$ is defined on $I$ and $F$ is an antiderivative of $f$ on $I$.     * The rule states: $\int f(g(x))g'(x)dx = F(g(x)) + C$.

• Proof of Theorem 1.4:     * By the chain rule of differentiation: $\frac{d}{dx} F(g(x)) = F'(g(x))g'(x)$.     * Since $F'(x) = f(x)$, we substitute $g(x)$ to find $\frac{d}{dx} F(g(x)) = f(g(x))g'(x)$.     * Therefore, $\int f(g(x))g'(x) dx = F(g(x)) + C$.

STEPS FOR SOLVING INTEGRALS OF COMPOSITE FUNCTIONS

• STEP 1: Determine the outer function $f(x)$ and the inner function $g(x)$.
• STEP 2: Find an antiderivative of the outer function $f(x)$ with the constant of integration equal to $0$.
• STEP 3: Explicitly solve for the composite expression $f(g(x))$ and the derivative of the inner function $g'(x)$.
• STEP 4: Rewrite the original integral so it contains the form $\int f(g(x))g'(x)dx$.
• STEP 5: Apply Theorem 1.4 to find the final result.

EXAMPLES: SOLVING COMPOSITE FUNCTION INTEGRALS

• Example 1: $\int 20x(x^2 + 1)^9 dx$     * STEP 1: Let $f(x) = x^9$ and $g(x) = x^2 + 1$.     * STEP 2: Antiderivative of $f(x)$ is $F(x) = \int x^9 dx = \frac{1}{10}x^{10}$.     * STEP 3: $f(g(x)) = f(x^2 + 1) = (x^2 + 1)^9$; and $g'(x) = \frac{d}{dx}(x^2 + 1) = 2x$.     * STEP 4: $\int 20x(x^2 + 1)^9 dx = 10 \int (x^2 + 1)^9 (2x) dx = 10 \int f(g(x))g'(x)dx$.     * STEP 5: $10 [F(g(x)) + C_1] = 10 [\frac{1}{10}(x^2 + 1)^{10}] + C = (x^2 + 1)^{10} + C$.

• Example 2: $\int (-3 \sin(3x - 2)) dx$     * STEP 1: Let $f(x) = \sin(x)$ and $g(x) = 3x - 2$.     * STEP 2: Antiderivative of $f(x)$ is $F(x) = \int \sin(x) dx = -\cos(x)$.     * STEP 3: $f(g(x)) = \sin(3x - 2)$; and $g'(x) = \frac{d}{dx}(3x - 2) = 3$.     * STEP 4: $\int (-3 \sin(3x - 2)) dx = -\int \sin(3x - 2)(3) dx = -\int f(g(x))g'(x)dx$.     * STEP 5: $-[F(g(x)) + C_1] = -(-\cos(3x - 2)) + C = \cos(3x - 2) + C.\n\n* **Example 3: \int e^{5x} dx**\n    * **STEP 1:** Let f(x) = e^x$and$g(x) = 5x.\n    * **STEP 2:** Antiderivative of f(x)$is$F(x) = \int e^x dx = e^x.\n    * **STEP 3:** f(g(x)) = e^{5x}$; and$g'(x) = \frac{d}{dx}(5x) = 5.\n    * **STEP 4:** \int e^{5x} dx = \frac{1}{5} \int e^{5x}(5) dx = \frac{1}{5} \int f(g(x))g'(x)dx.\n    * **STEP 5:** \frac{1}{5} [F(g(x)) + C_1] = \frac{1}{5}e^{5x} + C.\n\n# DIFFERENTIAL EQUATIONS\n\n* **Definition 1.3: Differential Equation**\n    * An equation containing a function and its derivatives, or just derivatives, is called a differential equation.\n    * **Examples:**\n        1. \frac{df}{dx} = 2x\n        2. \frac{df}{dx} = \frac{2x^2}{3f^3}\n        3. 3 \frac{d^2 f}{dx^2} + 2 \frac{df}{dx} = f\n\n* **Definition of a Solution:**\n    * A solution to a differential equation is a function f(x)$that satisfies the equation for all possible values of the variable$x.\n\n# SOLVING DIFFERENTIAL EQUATIONS\n\n* **Problem 1: Solve 3 \frac{df}{dx} = 2x**\n    * **Step 1:** Separate terms in f$from terms in$x$:$3 df = 2x dx.\n    * **Step 2:** Integrate both sides: 3 \int df = 2 \int x dx.\n    * **Step 3:** Solve the left side: 3 \int df = 3f + C_1.\n    * **Step 4:** Solve the right side: 2 \int x dx = x^2 + C_2.\n    * **Final Solution:** 3f = x^2 + C$or$f(x) = \frac{x^2 + C}{2}.\n\n* **Problem 2: Solve \frac{df}{dx} = \frac{2 + 3x^2}{f - 1}**\n    * **Step 1:** Separate terms in f$from terms in$x$:$(f - 1) df = (2 + 3x^2) dx.\n    * **Step 2:** Integrate both sides: \int (f - 1) df = \int (2 + 3x^2) dx.\n    * **Step 3:** Solve the left side: \int (f - 1) df = \int f df - \int df = \frac{1}{2}f^2 - f + C_1.\n    * **Step 4:** Solve the right side: \int (2 + 3x^2) dx = 2 \int dx + 3 \int x^2 dx = 2x + x^3 + C_2.\n    * **Final Solution:** \frac{1}{2}f^2 - f = 2x + x^3 + C$$.

REFERENCES

• Leithold, L. (1996), The calculus 7. Singapore: Addison Wesley Longman, Inc.
• Minton, R. & Smith, R. (2016). Basic calculus. Philippines: McGraw Hill Education.