3.2 Indefinite Integral
Indefinite integration is the reverse process of differentiation. When a rate of change or gradient function \(\frac{dy}{dx}\) is known, integration allows us to find the original function \(y = f(x)\). Because differentiating any constant yields zero, integrating a function produces a family of curves represented with an arbitrary constant of integration, \(c\). By using given boundary conditions or coordinates of a point on the curve, the exact value of \(c\) can be determined.
Learning Objectives
- State and use the basic indefinite integral formulas for a constant \(a\) and an algebraic function \(ax^n\) where \(n \neq -1\).
- Apply the addition and subtraction rules of integration for algebraic terms.
- Determine the indefinite integral for composite linear functions of the form \((ax + b)^n\), where \(a\) and \(b\) are constants and \(n \neq -1\), using substitution or standard formula.
- Find the constant of integration, \(c\), and determine the specific equation of a curve from its gradient function \(\frac{dy}{dx}\) and a given point \((x, y)\).
Flowchart: Steps to Integrate an Algebraic Term and Find the Curve Equation
A clear sequential visual aid for integrating power expressions and finding the exact curve equation using given coordinates.
Integrating \(ax^n\)
Add \(1\) to the power of \(x\).
Divide by the new power.
Add the constant of integration, \(+ c\).
Finding the Curve Equation
\(\dfrac{dy}{dx} \xrightarrow{\int \dots dx} y = F(x) + c \xrightarrow{\text{substitute }(x,y)} \text{Find } c \rightarrow \text{Final Curve Equation}\)
Short Explanation
Indefinite integration reverses differentiation to recover an original function from its rate of change.
Basic Indefinite Integral Formulas
\(\int a \, dx = ax + c\)
\(\int ax^n \, dx = \dfrac{ax^{n+1}}{n+1} + c\), where \(n \neq -1\).
Constant of Integration, \(c\)
Differentiating \(y = 5x\), \(y = 5x + 2\), and \(y = 5x - 3\) all yield \(\dfrac{dy}{dx} = 5\). Therefore, \(\int 5 \, dx = 5x + c\).
Properties of Indefinite Integrals
Constant factor rule: \(\int ax^n \, dx = a \int x^n \, dx\).
Addition and subtraction rule: \(\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx\).
Composite Linear Functions
\(\int (ax + b)^n \, dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + c\), for \(n \neq -1\).
This shortcut is valid only when the inside expression is linear.
| Type | Formula | Condition |
|---|---|---|
| Constant | \(\int a \, dx = ax + c\) | \(a\) is a constant |
| Power function | \(\int ax^n \, dx = \dfrac{ax^{n+1}}{n+1} + c\) | \(n \neq -1\) |
| Linear composite | \(\int (ax + b)^n \, dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + c\) | \(n \neq -1\), inside term is linear |
Steps to Integrate \(ax^n\) with Respect to \(x\)
Add \(1\) to the index of \(x\) (new index: \(n + 1\)).
Divide the term by the new index (\(n + 1\)).
Add the constant of integration, \(c\).
Steps to Find the Equation of a Curve from a Gradient Function
Integrate \(f'(x)\) with respect to \(x\), writing the expression with \(+ c\).
Substitute the given coordinates of a point \((x, y)\) that lies on the curve into the equation.
Solve the resulting linear equation for the constant of integration, \(c\).
Rewrite the equation of the curve with the calculated numerical value of \(c\).
Indefinite integral of a power
\(\int ax^n \, dx = \dfrac{ax^{n+1}}{n+1} + c\)Valid for \(n \neq -1\).
Indefinite integral of a linear composite function
\(\int (ax + b)^n \, dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + c\)Valid for \(n \neq -1\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Find \(\int \left(3x^2 - \dfrac{2}{x^2}\right) dx\).
Find the indefinite integral \(\int (3x - 5)^4 \, dx\).
The gradient function of a curve is \(\dfrac{dy}{dx} = 4x - 2\). Find the equation of the curve if it passes through the point \((-1, 7)\).
Common Mistakes
- Forgetting to write the constant of integration, \(+ c\), after evaluating an indefinite integral.
- Forgetting to divide by the coefficient \(a\) when integrating linear composite functions of the form \((ax + b)^n\), resulting in \(\dfrac{(ax+b)^{n+1}}{n+1} + c\) instead of \(\dfrac{(ax+b)^{n+1}}{a(n+1)} + c\).
- Attempting to integrate products or quotients directly without expanding or dividing first, for example integrating \(\int (x-2)(x+6) \, dx\) as the product of two separate integrals.
- Applying the formula \(\int (ax+b)^n \, dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + c\) to non-linear inner functions such as \(\int (3x^2 + 1)^3 \, dx\).
Concept Misunderstandings
Adding an independent constant of integration for every individual integrated term, for example writing \(x^3 + c_1 + 2x + c_2\).
Since the sum or difference of arbitrary constants is simply another arbitrary constant, only one single \(+ c\) is required for the entire integrated expression.
Believing that the power formula applies when \(n = -1\).
The formula \(\int ax^n \, dx = \dfrac{ax^{n+1}}{n+1} + c\) explicitly requires \(n \neq -1\), because when \(n = -1\), the denominator \(n + 1 = 0\), which is undefined.
Summary
- Indefinite integral of a constant: \(\int a \, dx = ax + c\).
- Indefinite integral of a power: \(\int ax^n \, dx = \dfrac{ax^{n+1}}{n+1} + c\) for \(n \neq -1\).
- Indefinite integral of a linear composite function: \(\int (ax + b)^n \, dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + c\) for \(n \neq -1\).
- Curve reconstruction: Given \(\dfrac{dy}{dx} = f'(x)\), integrate to find \(y = \int f'(x) \, dx = F(x) + c\), and substitute a known point \((x, y)\) to determine the exact value of \(c\).
Short Activity
Objective: Test your understanding of indefinite integration formulas and curve reconstruction.
A. Objective Quiz
1 What is the result of \(\int (2 - 3x)^4 \, dx\)?
2 A curve has a gradient function \(\dfrac{dy}{dx} = 6x^2 - 4\) and passes through \((1, 3)\). What is the constant of integration, \(c\)?
B. Fill in the Blanks
3 The indefinite integral of a constant \(a\) with respect to \(x\) is given by \(\int a \, dx = \text{_____}\).
4 In the indefinite integral formula \(\int ax^n \, dx = \dfrac{ax^{n+1}}{n+1} + c\), the index \(n\) cannot be equal to \(\text{_____}\).
C. Matching / Drag and Drop
Match each indefinite integral to its correct integrated algebraic result. On a phone, tap a card first, then tap the matching box.
