Integrating Expressions Revision Notes for Leaving Cert Mathematics

Integrating Expressions

Integrating powers of $x$ is a fundamental skill in calculus, and it involves finding the antiderivative (or integral) of functions of the form $x^n,$ where n is a real number. Here's a step-by-step guide on how to integrate powers of $x$ .

1. The General Rule for Integrating Powers of $x$ :

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The integral of $x^n$ with respect to $x$ is given by: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1$

Here, $n$ is any real number except $n = -1$ .
C is the constant of integration, which is included because indefinite integrals represent a family of functions.

2. Special Case: n = -1

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When $n =$ -1, the integral is: $\int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln|x| + C$

This is a special case because the formula $\frac{x^{n+1}}{n+1}$ does not work when $n$ = -1, as it would involve division by zero.

3. Examples of Integrating Powers of $x$ :

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

Here, $n$ = $2$ .
Apply the general rule: $\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$

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Example 2: Integrate $\int x^{-3} \, dx$

Here, $n$ = - $3$ .
Apply the general rule: $\int x^{-3} \, dx = \frac{x^{-3+1}}{-3+1} + C = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$

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Example 3: Integrate $\int \frac{1}{x^4} \, dx$

Rewrite $\frac{1}{x^4} as x^{-4}.$
Apply the general rule: $\int x^{-4} \, dx = \frac{x^{-4+1}}{-4+1} + C = \frac{x^{-3}}{-3} + C = -\frac{1}{3x^3} + C$

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Example 4: Integrate $\int \frac{1}{x} \, dx$

Recognise that $\frac{1}{x} = x^{-1}.$
Use the special case formula: $\int \frac{1}{x} \, dx = \ln|x| + C$

4. Applying the Rule in Definite Integrals:

To find the definite integral of $x^n$ from a to $b$ , you evaluate the antiderivative at the upper and lower limits and subtract:

$\int_a^b x^n \, dx = \left[ \frac{x^{n+1}}{n+1} \right]_a^b = \frac{b^{n+1}}{n+1} - \frac{a^{n+1}}{n+1}, \quad \text{for } n \neq -1$

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Example: Compute $\int_1^4 x^2 \, dx$

Find the antiderivative: $\int x^2 \, dx = \frac{x^3}{3} + C$
Evaluate from $1$ to $4$ : $\int_1^4 x^2 \, dx = \left[\frac{x^3}{3}\right]_1^4 = \frac{4^3}{3} - \frac{1^3}{3} = \frac{64}{3} - \frac{1}{3} = \frac{63}{3} = 21$

5. Summary of Key Points:

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The general rule for integrating $x^n is \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, provided\ n \neq -1.$
For
the integral is $\int x^{-1} \, dx = \ln|x| + C.$
The constant of integration $C$ is always included in indefinite integrals.
Definite integrals are computed by evaluating the antiderivative at the upper and lower limits.

Integration

The Antiderivative:

The antiderivative is what we find when reversing the process of differentiation.

The process of reversing differentiation is called integration.

Differentiation:

Multiply by the power of $x$ .
Subtract $1$ from the power.

Integration:

Add $1$ to the power.
Divide by the new power. Notation:

The symbol for integration is $\int$ .

$\int \quad \text{expression to integrate} \quad dx \quad \text{variable to integrate}$

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Examples:

$\int (x^2 + 2x) \, dx = \frac{x^3}{3} + \frac{2x^2}{2} + c$
Find $\frac{d}{dx} (x^3) = 3x^2$ Find $\int (3x^2) \, dx = \frac{3x^3}{3} + c = x^3 + c$
Find $\frac{d}{dx} (x^2 + 1) = 2x$ $\int (2x) \, dx = \frac{2x^2}{2} + c = x^2 + c$

Note: All integrations have a constant $+c$ .

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Examples:

$\int (5 \sqrt{x} + 2) \, dx$

= \int (5x^{\frac{1}{2}} + 2) \, dx

= 5x^{\frac{3}{2}}\times{\frac{2}{3}} + 2x + c

= \frac{10}{3} x^{\frac{3}{2}} + 2x + c

$\int \left(\frac{5}{2\sqrt{x}}\right) \, dx$

= \int \left(\frac{5}{2} x^{-\frac{1}{2}}\right) \, dx

= \frac{5}{\cancel2} x^{\frac{1}{2}}\times{\frac{\cancel2}{1}} + c

= \frac{10}{2} x^{\frac{1}{2}} + c

= 5x^{\frac{1}{2}} + c

Integrating Expressions (Leaving Cert Mathematics): Revision Notes