Rules of Integration Revision Notes for Leaving Cert Applied Maths

Rules of Integration

What is integration?

Integration is the mathematical process that reverses differentiation. When we differentiate a function, we find its rate of change. When we integrate, we're essentially working backwards to find the original function from its rate of change.

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Think of it this way: if differentiation asks "what's the slope?", then integration asks "what function would give us this slope when differentiated?"

The power rule for integration

The most fundamental rule of integration mirrors the power rule for differentiation, but works in reverse. While differentiation involved multiplying by the power and reducing it by one, integration requires us to add one to the power and then divide by this new power.

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The Power Rule for Integration:

$\int x^n , dx = \frac{x^{n+1}}{n+1} + c$

This is the most essential formula in integration - master this first!

Let's break down what each symbol means:

$\int$ is the integral symbol (like a stretched "S" for "sum")
$dx$ tells us we're integrating with respect to the variable x
$c$ is the constant of integration (explained below)

Why do we need the constant of integration?

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Why the Constant of Integration is Essential

The constant of integration might seem puzzling at first, but it's essential for a very important reason. Consider two functions: $f(x) = x^2 + 5$ and $g(x) = x^2 - 3$ .

When we differentiate both functions:

$f'(x) = 2x$
$g'(x) = 2x$

Both derivatives are identical, even though the original functions were different! This happens because the derivative of any constant is zero.

When we integrate, we need to account for this "lost" constant. That's why we add " $+c$ " to every indefinite integral - it represents any possible constant that could have been in the original function.

Working with the power rule

Let's see how the power rule works with some practical examples:

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Worked Example 1: $\int x^2 , dx$

Following our rule: add 1 to the power $(2+1=3)$ , then divide by the new power.

$\int x^2 , dx = \frac{x^3}{3} + c$

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Worked Example 2: $\int 4x^3 + 3 , dx$

We can separate this and handle each term:

For $4x^3$ : the coefficient 4 stays in front, $x^3$ becomes $\frac{x^4}{4}$
For the constant 3: constants integrate to give the constant times $x$

$\int 4x^3 + 3 , dx = x^4 + 3x + c$

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Worked Example 3: $\int \frac{1}{x^3} , dx$

Step 1: Rewrite using indices: $\frac{1}{x^3} = x^{-3}$

Step 2: Apply the rule: $\frac{x^{-3+1}}{-3+1} = \frac{x^{-2}}{-2} = -\frac{1}{2x^2} + c$

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

Logarithmic integration

One special case that doesn't follow the standard power rule is when we integrate $\frac{1}{x}$ . This is because applying the power rule to $x^{-1}$ would involve dividing by zero, which is undefined.

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Logarithmic Integration Rules:

$\int \frac{1}{x} , dx = \ln|x| + c$

$\int \frac{1}{ax+b} , dx = \frac{1}{a}\ln|ax+b| + c$

The first formula appears in standard mathematical tables, while the second (more general) form often doesn't, so it's worth memorising.

Integration of exponential and trigonometric functions

Just as we used tables for differentiation of these special functions, we use similar approaches for integration. Many of these standard integrals can be found in mathematical reference tables.

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Key Exponential and Trigonometric Integration Formulas:

$\int e^{ax} , dx = \frac{1}{a}e^{ax} + c$
$\int \sin(axe) , dx = -\frac{1}{a}\cos(axe) + c$
$\int \cos(axe) , dx = \frac{1}{a}\sin(axe) + c$

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The formulas for $\sin(axe)$ and $\cos(axe)$ shown above are typically NOT found in standard tables - they're more general forms that you should memorise.

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Worked Example: $\int \sin(5x) , dx$

Using our formula with $a = 5$ :

$\int \sin(5x) , dx = -\frac{1}{5}\cos(5x) + c$

Inverse trigonometric integrals

Some more advanced integration problems result in inverse trigonometric functions. These special cases are worth knowing for more complex problems.

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Inverse Trigonometric Integration Formulas:

$\int \frac{1}{\sqrt{a^2-x^2}} , dx = \sin^{-1}\left(\frac{x}{a}\right) + c$

$\int \frac{1}{x^2+a^2} , dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + c$

These formulas are particularly useful when dealing with rational functions involving square roots or squared terms.

Definite integrals

Sometimes we need to find the exact numerical value of an integral between two specific points. These are called definite integrals and use limits of integration.

For definite integrals, we don't need the constant of integration because we're finding a specific numerical answer.

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The Fundamental Theorem of Calculus:

$\int\_a^b f(x) , dx = F(b) - F(a)$

Where $F(x)$ is the antiderivative of $f(x)$ , and $\[a,b]$ represents the limits from $a$ to $b$ .

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Worked Example: $\int\_{-2}^{2} x(x + 4) , dx$

Step 1: Expand and integrate $\int (x^2 + 4x) , dx = \frac{x^3}{3} + 2x^2$

Step 2: Apply the limits $F(2) - F(-2) = \left\[\frac{8}{3} + 8\right] - \left\[\frac{(-8)}{3} + 8\right] = \frac{16}{3}$

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Exam Tips:

Always add +c for indefinite integrals, but never for definite integrals
Check your answer by differentiating - you should get back to the original function
Rewrite expressions using indices when dealing with roots or fractions
Use tables wisely - know which formulas are in tables and which ones you need to memorise
For definite integrals, always subtract F(lower limit) from F(upper limit)

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Key Points to Remember:

Integration is the reverse process of differentiation - it finds the original function from its derivative
The power rule for integration: add one to the power, then divide by the new power
Always include the constant of integration (+c) for indefinite integrals
Special cases like $\frac{1}{x}$ integrate to natural logarithms, not following the standard power rule
Many trigonometric and exponential integrals can be found in mathematical tables, but some key formulas should be memorised

Rules of Integration (Leaving Cert Applied Maths): Revision Notes