Integrating Powers of x

The power rule for integration and techniques for simplifying before integrating

SPEC 8.2 • INTEGRATION

The Power Rule for Integration

Power Rule

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c where n \neq = - 1

Method

Add one to the power, then divide by the new power.

FunctionIntegral

x^{2}

\frac{x ^{3}}{3} + c

x^{5}

\frac{x ^{6}}{6} + c

x^{- 3}

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

x^{1/2}

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

Why the Power Rule Works

The power rule is the reverse of differentiation.

Differentiation

\frac{d}{d x} (x^{n}) = n x^{n - 1}

Multiply by power, subtract one

Integration

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c

Add one to power, divide by new power

Verification

Check: If

\int x^{3} d x = \frac{x ^{4}}{4} + c

, then

\frac{d}{d x} (\frac{x ^{4}}{4} + c) = \frac{4 x ^{3}}{4} = x^{3}

✓

Constants and Constant Multiples

Integrating Constants

\int k d x = k x + c

Examples:

\int 5 d x = 5 x + c

\int (- 3) d x = - 3 x + c

Constant Multiples

\int k f (x) d x = k \int f (x) d x

Examples:

\int 6 x^{2} d x = 6 \cdot \frac{x ^{3}}{3} = 2 x^{3} + c

\int 10 x^{4} d x = 10 \cdot \frac{x ^{5}}{5} = 2 x^{5} + c

Direct Method

For

\int a x^{n} d x

, the result is

\frac{a \cdot x ^{n + 1}}{n + 1} + c

— integrate directly!

Integrating Sums and Differences

Integrate term by term:

Rule

\int [f (x) \pm g (x)] d x = \int f (x) d x \pm \int g (x) d x

Example:

\int (3 x^{2} + 4 x - 5) d x

= \int 3 x^{2} d x + \int 4 x d x - \int 5 d x

= x^{3} + 2 x^{2} - 5 x + c

Important

Only one constant of integration is needed at the end, not one for each term.

Simplifying Before Integrating

You often need to rewrite expressions before the power rule can be applied:

TechniqueOriginalRewrite as

Expand brackets

(x + 3)^{2}

x^{2} + 6 x + 9

Roots → fractional powers

x, 3 x

x^{1/2}, x^{1/3}

Fractions → negative powers

\frac{1}{x ^{2}}, \frac{1}{x}

x^{- 2}, x^{- 1/2}

Split fractions

\frac{x ^{3} + 2 x}{x}

x^{2} + 2

Warning

\int (x + 3)^{2} d x \neq = \frac{( x + 3 ) ^{3}}{3} + c

— this is WRONG! You must expand first.

The Exception: n = −1

The power rule fails when

n = - 1

:

Problem

\int x^{- 1} d x = \frac{x ^{0}}{0} + c = \frac{1}{0} + c — undefined!

This means we cannot use the power rule for

\int \frac{1}{x} d x

or

\int x^{- 1} d x

.

The Answer

\int \frac{1}{x} d x = ln ∣ x ∣ + c

This will be covered in the lesson on integrating logarithmic functions.

Remember

Always check: if

n = - 1

, you cannot use the power rule!

Worked Example01

Find $\int (4 x^{3} - 6 x^{2} + 5) d x$

1

Integrate each term using the power rule:

•

\int 4 x^{3} d x = 4 \cdot \frac{x ^{4}}{4} = x^{4}

•

\int 6 x^{2} d x = 6 \cdot \frac{x ^{3}}{3} = 2 x^{3}

•

\int 5 d x = 5 x

2

Combine with

+ c

:

Answer

x^{4} - 2 x^{3} + 5 x + c

Worked Example02

Find $\int (x + \frac{3}{x ^{2}}) d x$

1

Rewrite using powers:

•

x = x^{1/2}

and

\frac{3}{x ^{2}} = 3 x^{- 2}

2

Integrate each term:

•

\int x^{1/2} d x = \frac{x ^{3/2}}{3/2} = \frac{2}{3} x^{3/2}

•

\int 3 x^{- 2} d x = 3 \cdot \frac{x ^{- 1}}{- 1} = - \frac{3}{x}

Answer

\frac{2}{3} x^{3/2} - \frac{3}{x} + c

or

\frac{2}{3} x^{3} - \frac{3}{x} + c

Worked Example03

Find $\int x (x - 3)^{2} d x$

1

Expand

(x - 3)^{2}

first:

•

(x - 3)^{2} = x^{2} - 6 x + 9

2

Multiply by

x

:

•

x (x^{2} - 6 x + 9) = x^{3} - 6 x^{2} + 9 x

3

Integrate:

•

\int (x^{3} - 6 x^{2} + 9 x) d x = \frac{x ^{4}}{4} - 2 x^{3} + \frac{9 x ^{2}}{2} + c

Answer

\frac{x ^{4}}{4} - 2 x^{3} + \frac{9 x ^{2}}{2} + c

Worked Example04

Find $\int \frac{x ^{4} - 3 x ^{2} + 2}{x ^{2}} d x$

1

Split the fraction by dividing each term by

x^{2}

:

•

\frac{x ^{4} - 3 x ^{2} + 2}{x ^{2}} = \frac{x ^{4}}{x ^{2}} - \frac{3 x ^{2}}{x ^{2}} + \frac{2}{x ^{2}} = x^{2} - 3 + 2 x^{- 2}

2

Integrate each term:

•

\int (x^{2} - 3 + 2 x^{- 2}) d x = \frac{x ^{3}}{3} - 3 x + 2 \cdot \frac{x ^{- 1}}{- 1} + c

Answer

\frac{x ^{3}}{3} - 3 x - \frac{2}{x} + c

Worked Example05

Find $\int (3 x + \frac{2}{x ^{4}}) d x$

1

Rewrite using powers:

•

3 x = x^{1/3}

and

\frac{2}{x ^{4}} = 2 x^{- 4}

2

Integrate each term:

•

\int x^{1/3} d x = \frac{x ^{4/3}}{4/3} = \frac{3}{4} x^{4/3}

•

\int 2 x^{- 4} d x = 2 \cdot \frac{x ^{- 3}}{- 3} = - \frac{2}{3} x^{- 3} = - \frac{2}{3 x ^{3}}

Answer

\frac{3}{4} x^{4/3} - \frac{2}{3 x ^{3}} + c

or

\frac{3}{4} 3 x^{4} - \frac{2}{3 x ^{3}} + c

Worked Example06

Find $\int (2 x + 1) (x - 4) d x$

1

Expand the brackets:

•

(2 x + 1) (x - 4) = 2 x^{2} - 8 x + x - 4 = 2 x^{2} - 7 x - 4

2

Integrate each term:

•

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

•

\int 7 x d x = \frac{7 x ^{2}}{2}

•

\int 4 d x = 4 x

Answer

\frac{2 x ^{3}}{3} - \frac{7 x ^{2}}{2} - 4 x + c

Worked Example07

Find $\int (x^{2} x - \frac{4}{x}) d x$

1

Rewrite using index laws:

•

x^{2} x = x^{2} \cdot x^{1/2} = x^{5/2}

•

\frac{4}{x} = 4 x^{- 1/2}

2

Integrate each term:

•

\int x^{5/2} d x = \frac{x ^{7/2}}{7/2} = \frac{2}{7} x^{7/2}

•

\int 4 x^{- 1/2} d x = 4 \cdot \frac{x ^{1/2}}{1/2} = 8 x^{1/2} = 8 x

Answer

\frac{2}{7} x^{7/2} - 8 x + c

or

\frac{2}{7} x^{7} - 8 x + c

Practice

Question 1: Find

\int (5 x^{4} - 3 x^{2} + 7 x - 2) d x

1

Integrate each term:

•

\int 5 x^{4} d x = x^{5}

,

\int 3 x^{2} d x = x^{3}

,

\int 7 x d x = \frac{7 x ^{2}}{2}

,

\int 2 d x = 2 x

Answer

x^{5} - x^{3} + \frac{7 x ^{2}}{2} - 2 x + c

Practice

Question 2: Find

\int (\frac{4}{x} - \frac{1}{x ^{3}}) d x

1

Rewrite:

\frac{4}{x} = 4 x^{- 1/2}

and

\frac{1}{x ^{3}} = x^{- 3}

2

Integrate:

•

\int 4 x^{- 1/2} d x = 4 \cdot \frac{x ^{1/2}}{1/2} = 8 x

•

\int x^{- 3} d x = \frac{x ^{- 2}}{- 2} = - \frac{1}{2 x ^{2}}

Answer

8 x + \frac{1}{2 x ^{2}} + c

Practice

Question 3: Find

\int (3 x - 2) (x + 5) d x

1

Expand the brackets:

•

(3 x - 2) (x + 5) = 3 x^{2} + 15 x - 2 x - 10 = 3 x^{2} + 13 x - 10

2

Integrate:

•

\int (3 x^{2} + 13 x - 10) d x = x^{3} + \frac{13 x ^{2}}{2} - 10 x + c

Answer

x^{3} + \frac{13 x ^{2}}{2} - 10 x + c

Summary

Key Formulas

RuleFormula

Power Rule

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c (n \neq = - 1)

Constants

\int k d x = k x + c

Constant Multiples

\int k f (x) d x = k \int f (x) d x

Sums/Differences

\int [f (x) \pm g (x)] d x = \int f (x) d x \pm \int g (x) d x

Before Integrating

Expand any brackets

Rewrite roots as fractional powers

Rewrite fractions as negative powers

Split fractions into separate terms