FAD1014 L1-L2 — Integration (Anti-Derivative)

Comprehensive lecture notes covering the fundamentals of integration as the reverse process of differentiation. Lecturer: En Jedzry Fadzlin B Jalaluddin (BP217). Material adapted from J Merrill 2009 and MZMK 2011.

1.1 — Antiderivatives

Practical Motivation

Integrals appear in many practical situations. For simple rectangular shapes, we can use basic geometry, but for irregular or curved shapes (e.g., an oval swimming pool with a rounded bottom), precision engineering requires exact and rigorous values — these call for integrals.

We have been solving situations dealing with total amounts of quantities. Derivatives deal with the rate of change of those quantities. Since it is not always possible to find functions that deal with the total amount directly, we need antidifferentiation.

Kinematics Connection

The relationship between distance, velocity, and acceleration illustrates differentiation vs integration:

Differentiation: $$s(t) \xrightarrow{\frac{ds}{dt}} v(t) \xrightarrow{\frac{dv}{dt}} a(t)$$

Integration (reverse): $$a(t) \xleftarrow{\int a(t),dt} v(t) \xleftarrow{\int v(t),dt} s(t)$$

flowchart LR
    subgraph Diff["Differentiation"]
        D1[s(t)] -->|"d/dt"| D2[v(t)]
        D2 -->|"d/dt"| D3[a(t)]
    end
    subgraph Int["Integration"]
        I3[a(t)] -->|"∫"| I2[v(t)]
        I2 -->|"∫"| I1[s(t)]
    end
    Diff -.->|"Reverse process"| Int

Definition

Antiderivative: If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

Examples:

  • If $F(x) = 10x$, then $F'(x) = 10$. So $F(x)$ is the antiderivative of $f(x) = 10$.
  • If $F(x) = x^2$, then $F'(x) = 2x$. So $F(x)$ is the antiderivative of $f(x) = 2x$.
  • The antiderivative of $f(x) = 5x^4$ is $x^5$ (work backwards from the derivative).

The Constant of Integration

$F(x) = x^2$ is not the only function whose derivative is $f(x) = 2x$:

  • $G(x) = x^2 + 2$ has derivative $2x$
  • $H(x) = x^2 - 7$ has derivative $2x$

For any real number $C$, the function $F(x) = x^2 + C$ has $f(x) = 2x$ as an antiderivative. There is a whole family of functions having $2x$ as a derivative; this family differs only by a constant. Graphically, these are vertical shifts of the same curve with identical tangent slopes at any given $x$.

Theorem: If $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$ on an interval, then there is a constant $C$ such that $F(x) - G(x) = C$. Two antiderivatives of a function can differ only by a constant. The arbitrary real number $C$ is called an integration constant.

Indefinite Integral Notation

The family of all antiderivatives of $f$ is indicated by: $$\int f(x),dx$$

  • $\int$ = integral sign
  • $f(x)$ = integrand
  • $dx$ = indicates integration with respect to $x$

Indefinite Integral: If $F'(x) = f(x)$, then $$\int f(x),dx = F(x) + C$$ for any real number $C$. This is the most general antiderivative of $f$.

Variable of Integration

The symbol after $d$ indicates the variable of integration; other letters are treated as constants.

Example 1.1.1:

  • $\displaystyle\int 2ax,dx = a(2x),\text{? No: } \int 2ax,dx = ax^2 + C$
    Here $a$ is treated as a constant and $x$ as the variable.
  • $\displaystyle\int 2ax,da = a^2x + C = xa^2 + C$
    Here $x$ is treated as the constant.

Finding the Antiderivative (Power Rule)

Finding the antiderivative is the reverse of finding the derivative. Therefore, the rules for derivatives lead to rules for antiderivatives.

Example: $$\frac{d}{dx}x^5 = 5x^4 \quad\Longrightarrow\quad \int 5x^4,dx = x^5 + C$$

Example 1.1.2:

  • $\displaystyle\int x^{-1/2},dx$
  • $\displaystyle\int 3,dx = 3x + C$
  • $\displaystyle\int \frac{1}{x^3},dx$

Sigma Notation, Riemann Sums, and the Definite Integral

Sigma Notation

The summation symbol $\displaystyle\sum_{k=1}^{n} a_k$ denotes the sum of terms $a_k$ from $k=1$ to $k=n$.

Examples:

Sigma Notation Expanded Value
$\sum_{k=1}^{5} k$ $1+2+3+4+5$ $15$
$\sum_{k=1}^{3} (-1)^k k$ $(-1)^1(1)+(-1)^2(2)+(-1)^3(3)$ $-1+2-3 = -2$
$\sum_{k=1}^{2} \frac{k}{k+1}$ $\frac{1}{2}+\frac{2}{3}$ $\frac{7}{6}$
$\sum_{k=4}^{5} \frac{k^2}{k-1}$ $\frac{16}{3}+\frac{25}{4}$ $\frac{139}{12}$

Riemann Sum

If $f$ is continuous, the left Riemann sum with $n$ equal subdivisions over $[a,b]$ is: $$\sum_{k=0}^{n-1} f(x_k)\Delta x = \bigl[f(x_0)+f(x_1)+\dots+f(x_{n-1})\bigr]\Delta x$$ where $a=x_0 < x_1 < \dots < x_n=b$ and $\Delta x = \frac{b-a}{n}$.

The Definite Integral

$$\int_a^b f(x),dx = \lim_{n\to\infty} \sum_{k=0}^{n-1} f(x_k)\Delta x$$

  • $f$ = integrand
  • $a$, $b$ = limits of integration
  • $x$ = variable of integration

Example: Approximate $\displaystyle\int_0^2 x^2,dx$ using $n=10$ (left Riemann sum): $$\sum_{k=0}^{9} x_k^{,2}\left(\frac{1}{5}\right) = \Bigl[\left(\tfrac{1}{5}\right)^2 + \left(\tfrac{2}{5}\right)^2 + \dots + \left(\tfrac{9}{5}\right)^2\Bigr]\left(\frac{1}{5}\right) = 2.28$$

1.2 — Indefinite Integrals and Standard Integrals

1.2.1 Rules for Antiderivatives

Integrating a Constant: $$\int a,dx = ax + c \qquad (a, c \text{ are constants})$$

Power Rule: $$\int x^n,dx = \frac{x^{n+1}}{n+1} + C \qquad \text{for any real number } n \neq -1$$ (Add 1 to the exponent and divide by that number.)

You can always check your answers by taking the derivative!

Examples:

  • $\displaystyle\int t^3,dt = \frac{t^{3+1}}{3+1} = \frac{t^4}{4} + C$
  • $\displaystyle\int \frac{1}{t^2},dt = \int t^{-2},dt = \frac{t^{-1}}{-1} + C = -\frac{1}{t} + C$

You Do (practice):

  1. $\displaystyle\int \sqrt{u},du$
  2. $\displaystyle\int dx$

1.2.2 Constant Multiple and Sum/Difference

Basic Rules of Integration:

(a) Constant Multiple: If $k$ is a constant, $$\int k \cdot f(x),dx = k\int f(x),dx \qquad \text{for any real number } k$$

(b) Sum/Difference: For two functions $f(x)$ and $g(x)$, $$\int \bigl[f(x) \pm g(x)\bigr],dx = \int f(x),dx \pm \int g(x),dx$$

Example 1.2.1: $$\int 2v^3,dv = 2\int v^3,dv = 2\left(\frac{v^4}{4}\right) + C = \frac{v^4}{2} + C$$

You Do (practice):

  • $\displaystyle\int \frac{12}{z^5},dz$
  • $\displaystyle\int (3z^2 - 4z + 5),dz$

Example 1.2.2: $$\int \frac{x^2+1}{\sqrt{x}},dx$$

First, rewrite the integrand: $$= \int \left(\frac{x^2}{\sqrt{x}} + \frac{1}{\sqrt{x}}\right)dx = \int \left(x^{3/2} + x^{-1/2}\right)dx$$

Now integrate: $$= \frac{x^{5/2}}{\tfrac{5}{2}} + \frac{x^{1/2}}{\tfrac{1}{2}} + C = \frac{2}{5}x^{5/2} + 2x^{1/2} + C$$

1.2.3 Indefinite Integrals of Exponential Functions

Recall from differentiation:

  • If $f(x) = e^x$ then $f'(x) = e^x$
  • If $f(x) = a^x$ then $f'(x) = a^x(\ln a)$
  • If $f(x) = e^{kx}$ then $f'(x) = ke^{kx}$
  • If $f(x) = a^{kx}$ then $f'(x) = k(\ln a)a^{kx}$

This leads to the following integration formulas:

$$\int e^x,dx = e^x + C$$

$$\int e^{kx},dx = \frac{e^{kx}}{k} + C \qquad (k \neq 0)$$

$$\int a^x,dx = \frac{a^x}{\ln a} + C$$

$$\int a^{kx},dx = \frac{a^{kx}}{k(\ln a)} + C \qquad (k \neq 0)$$

General form: $$\int f'(x)e^{f(x)},dx = e^{f(x)} + c$$

Example 1.2.3:

  • $\displaystyle\int 9e^t,dt = 9\int e^t,dt = 9e^t + C$
  • $\displaystyle\int e^{9t},dt = \frac{e^{9t}}{9} + C$
  • $\displaystyle\int 3e^{\frac{5}{4}u},du = 3\left(\frac{e^{\frac{5}{4}u}}{\tfrac{5}{4}}\right) + C = 3\left(\frac{4}{5}\right)e^{\frac{5}{4}u} + C = \frac{12}{5}e^{\frac{5}{4}u} + C$

You Do (practice): $$\int 2^{-5x},dx$$ (Recall: $\int a^{kx},dx = \frac{a^{kx}}{k(\ln a)} + C$, $k \neq 0$)

1.2.4 Indefinite Integral of $x^{-1}$

The power rule does not apply when $n = -1$. Instead:

$$\int x^{-1},dx = \int \frac{1}{x},dx = \ln|x| + C$$

General form (produces log function): $$\int \frac{f'(x)}{f(x)},dx = \ln\bigl|f(x)\bigr| + c$$

Note: If $x$ takes on a negative value, then $\ln x$ is undefined. The absolute value sign keeps that from happening.

Example 1.2.4: $$\int \frac{4}{x},dx = 4\int \frac{1}{x},dx = 4\ln|x| + C$$

You Do (practice): $$\int \left(\frac{-5}{x} + e^{-2x}\right)dx$$

1.2.5 Indefinite Integrals of Trigonometric Functions

$$\int \sin x,dx = -\cos x + c$$

$$\int \cos x,dx = \sin x + c$$

$$\int \tan x,dx = \ln|\sec x| + c$$

$$\int \sec x,dx = \ln|\sec x + \tan x| + c$$

$$\int \cot x,dx = \ln|\sin x| + c$$

$$\int \sec^2 x,dx = \tan x + c$$

$$\int \csc x,dx = \ln|\csc x - \cot x| + c$$

$$\int \csc^2 x,dx = -\cot x + c$$

$$\int \sec x \tan x,dx = \sec x + c$$

$$\int \csc x \cot x,dx = -\csc x + c$$

Example 1.2.5: $$\int \left(3\cos x - 4\sin x + \frac{1}{x^2}\right)dx$$

$$= 3\int \cos x,dx - 4\int \sin x,dx + \int \frac{1}{x^2},dx$$

$$= 3\sin x + 4\cos x - \frac{1}{x} + C$$

Example 1.2.6: $$\int \left(3^x - 4e^x + \frac{7}{4x}\right)dx$$

$$= \int 3^x,dx - 4\int e^x,dx + \frac{7}{4}\int \frac{1}{x},dx$$

$$= \frac{3^x}{\ln 3} - 4e^x + \frac{7}{4}\ln|x| + C$$

Example 1.2.7: $$\int \frac{3x^2+4x+1}{x-1},dx$$

First, perform polynomial division / rewrite the integrand: $$= \int \left(3x + 7 + \frac{8}{x-1}\right)dx$$

$$= 3\int x,dx + 7\int dx + 8\int \frac{1}{x-1},dx$$

For the last term, let $u = x-1$, therefore $du = dx$: $$= 3\cdot\frac{x^2}{2} + 7x + 8\int \frac{1}{u},du$$

$$= \frac{3x^2}{2} + 7x + 8\ln(u) + C$$

$$= \frac{3x^2}{2} + 7x + 8\ln(x-1) + C$$

Key Equations Summary

Rule / Function Integral
Constant $\displaystyle\int a,dx = ax + C$
Power Rule ($n \neq -1$) $\displaystyle\int x^n,dx = \frac{x^{n+1}}{n+1} + C$
$x^{-1}$ $\displaystyle\int \frac{1}{x},dx = \ln|x| + C$
$e^x$ $\displaystyle\int e^x,dx = e^x + C$
$e^{kx}$ $\displaystyle\int e^{kx},dx = \frac{e^{kx}}{k} + C$
$a^x$ $\displaystyle\int a^x,dx = \frac{a^x}{\ln a} + C$
$a^{kx}$ $\displaystyle\int a^{kx},dx = \frac{a^{kx}}{k(\ln a)} + C$
General log $\displaystyle\int \frac{f'(x)}{f(x)},dx = \ln|f(x)| + C$
General exp $\displaystyle\int f'(x)e^{f(x)},dx = e^{f(x)} + C$
$\sin x$ $\displaystyle\int \sin x,dx = -\cos x + C$
$\cos x$ $\displaystyle\int \cos x,dx = \sin x + C$
$\tan x$ $\displaystyle\int \tan x,dx = \ln|\sec x| + C$
$\sec x$ $\displaystyle\int \sec x,dx = \ln|\sec x + \tan x| + C$
$\cot x$ $\displaystyle\int \cot x,dx = \ln|\sin x| + C$
$\sec^2 x$ $\displaystyle\int \sec^2 x,dx = \tan x + C$
$\csc x$ $\displaystyle\int \csc x,dx = \ln|\csc x - \cot x| + C$
$\csc^2 x$ $\displaystyle\int \csc^2 x,dx = -\cot x + C$
$\sec x \tan x$ $\displaystyle\int \sec x \tan x,dx = \sec x + C$
$\csc x \cot x$ $\displaystyle\int \csc x \cot x,dx = -\csc x + C$

Properties of Indefinite Integrals

  • Constant Multiple: $\displaystyle\int k \cdot f(x),dx = k\int f(x),dx$
  • Sum/Difference: $\displaystyle\int \bigl[f(x) \pm g(x)\bigr],dx = \int f(x),dx \pm \int g(x),dx$

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