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 $x$ :

infoNote

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

infoNote

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 $x$ :

infoNote

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

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

infoNote

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

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

infoNote

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

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

infoNote

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

Recognize 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:

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

infoNote

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:

infoNote

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 $\int$ .

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

infoNote

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 $+c$ .

infoNote

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 Simplified Revision Notes for Leaving Cert Mathematics

Integrating Expressions

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

2. Special Case: n = -1

3. Examples of Integrating Powers of $x$ :

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

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

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

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

4. Applying the Rule in Definite Integrals:

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

5. Summary of Key Points:

Integration

The Antiderivative:

Differentiation:

Integration:

Examples:

500K+ Students Use These Powerful Tools to Master Integrating Expressions For their Leaving Cert Exams.

Other Revision Notes related to Integrating Expressions you should explore

Integrating Expressions Simplified Revision Notes for Leaving Cert Mathematics

Integrating Expressions

1. The General Rule for Integrating Powers of xxx:

2. Special Case: n = -1

3. Examples of Integrating Powers of x xx:

Example 1: Integrate ∫x2 dx\int x^2 \, dx∫x2dx

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

Example 3: Integrate ∫1x4 dx\int \frac{1}{x^4} \, dx∫x41​dx

Example 4: Integrate ∫1x dx\int \frac{1}{x} \, dx∫x1​dx

4. Applying the Rule in Definite Integrals:

Example: Compute ∫14x2 dx\int_1^4 x^2 \, dx∫14​x2dx

5. Summary of Key Points:

Integration

The Antiderivative:

Differentiation:

Integration:

Examples:

500K+ Students Use These Powerful Tools to Master Integrating Expressions For their Leaving Cert Exams.

Other Revision Notes related to Integrating Expressions you should explore

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

3. Examples of Integrating Powers of $x$ :

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

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

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

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

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