What is Integration?

Integration is the inverse operation of differentiation. Just as subtraction undoes addition, integration undoes differentiation , this is why it is also called anti-differentiation or finding the anti-derivative.

If differentiation asks "what is the rate of change of this function?", integration asks "what function has this as its rate of change?"

Anti-derivative: A function $F (x)$ is called an anti-derivative of $f (x)$ if $F^{'} (x) = f (x)$ . That is, differentiating $F (x)$ gives back $f (x)$ .

Analogy

Think of integration like reversing a journey. If differentiation tells you your speed given your position, integration tells you your position given your speed. You're working backwards from what you know.

The Power Rule for Integration

For polynomial functions of the form $f (x) = a x^{n}$ , where $n$ is a real number and $n \neq = - 1$ , the anti-derivative is found using the power rule for integration:

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$

In plain terms: add one to the power, then divide by the new power. The rule extends to any polynomial by integrating term by term:

$\int (a x^{n} + b x^{m} + \dots) d x = \frac{a x ^{n + 1}}{n + 1} + \frac{b x ^{m + 1}}{m + 1} + \dots + C$

Warning

The rule does not apply when $n = - 1$ (i.e. for $\int \frac{1}{x} d x$ ). That case involves a logarithm and is beyond the SL syllabus.

Exam Tip

A quick memory trick: "Raise the power, divide by the new power." It's the exact reverse of the differentiation rule (multiply by the power, reduce the power by one).

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What is Integration?

If differentiation asks "what is the rate of change of this function?", integration asks "what function has this as its rate of change?"

Anti-derivative: A function $F (x)$ is called an anti-derivative of $f (x)$ if $F^{'} (x) = f (x)$ . That is, differentiating $F (x)$ gives back $f (x)$ .

Analogy

The Power Rule for Integration

For polynomial functions of the form $f (x) = a x^{n}$ , where $n$ is a real number and $n \neq = - 1$ , the anti-derivative is found using the power rule for integration:

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$

In plain terms: add one to the power, then divide by the new power. The rule extends to any polynomial by integrating term by term:

$\int (a x^{n} + b x^{m} + \dots) d x = \frac{a x ^{n + 1}}{n + 1} + \frac{b x ^{m + 1}}{m + 1} + \dots + C$

Warning

The rule does not apply when $n = - 1$ (i.e. for $\int \frac{1}{x} d x$ ). That case involves a logarithm and is beyond the SL syllabus.

Exam Tip

A quick memory trick: "Raise the power, divide by the new power." It's the exact reverse of the differentiation rule (multiply by the power, reduce the power by one).

Free preview

SL 5.5—Introduction to integration

What is Integration?

The Power Rule for Integration

9 more sections in this topic

SL 5.5—Introduction to integration

What is Integration?

The Power Rule for Integration

9 more sections in this topic