Integrating powers of x

5 minAS Level1.8.1

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can integrate sums of powers of $x$ , like $\displaystyle\int (2x^3 - 5x + 1)\,dx$ , and say why every answer ends in $+\,c$ .

Differentiation, in reverse

Differentiate $x^3$ and you get $3x^2$ .

Integration runs that backwards. It starts from $3x^2$ and asks: what did I differentiate to get this? Think of it as the undo button for differentiation.

What they ask

Find $\displaystyle\int (6x^2 - 4x + 3)\,dx$ .

The long S, $\displaystyle\int$ , means "integrate".
The $dx$ at the end says the letter is $x$ .
Your answer is a function of $x$ , ending in $+\,c$ .

Definition

Integrating a power of x

$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c \qquad (n \neq -1)$

$n$ is the power of $x$ ; it can be any number except $-1$ .
Add 1 to the power, then divide by the new power.
$c$ is the constant of integration: it can be any number.
It works for negative and fractional powers too. The only exception is $n = -1$ , because then you'd divide by 0.

Why + c?

$x^3$ , $\;x^3 + 7\;$ and $\;x^3 - 2\;$ all differentiate to $3x^2$ , because a constant differentiates to 0.

So when you integrate $3x^2$ , you can't know which constant was there. Writing $+\,c$ says "plus some number we don't know yet".

The method

Same four steps every time:

Write every term as a number times a power of $x$ . Expand brackets, split fractions, and turn roots into powers: $\sqrt{x} = x^{\frac12}$ and $\dfrac{1}{x^2} = x^{-2}$ .
Add 1 to each power.
Divide each term by its new power.
Add $+\,c$ . This is the step people forget, and it costs a mark.

Numbers in front come along for the ride, and you integrate a sum one term at a time.

Worked example: The easiest one: check it by differentiating

Find $\displaystyle\int 6x^2\,dx$ .

Have a go first?

Worked example: Several terms (the syllabus example)

Find $\displaystyle\int (2x^3 - 5x + 1)\,dx$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 3 questions above to continue.