How to Integrate a Polynomial

What this method is for

Integration is the reverse of differentiation. Where differentiation takes a curve to its gradient function, integration takes a gradient function back to the original curve.

Integrating a polynomial is the most basic and most common integration you will do in Add Math, and it underpins everything else in the chapter.

The tool is the power rule for integration: for each term, you increase the power by one and divide by the new power. Because a polynomial is just a sum of power terms, you simply apply the rule to every term in turn.

This method answers questions such as 'find $\int(6x^{2}-4x+3)\,dx$ ', and, crucially, questions that give you $\frac{dy}{dx}$ together with a point on the curve and ask you to find the equation $y$ . The constant of integration $c$ is where that point comes in.

When to reach for it

Reach for polynomial integration whenever you see the integral sign $\int$ around a sum of power terms, or whenever a question gives a derivative, $\frac{dy}{dx}$ , a gradient function, a velocity, or a rate, and asks you to recover the original quantity. Words like 'integrate', 'find $y$ in terms of $x$ ', or 'the curve for which $\frac{dy}{dx}=\dots$ ' are all signals.

If the question also gives a specific point the curve passes through, that is your cue to find the constant of integration rather than leaving $+c$ . For a definite integral, with limits written on the sign, you integrate the same way and then substitute the limits instead of adding $c$ .

Either way, the first move on any polynomial is the power rule, term by term.

The method, step by step

Power rule for integrationMust memorise

\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c,\quad n\neq -1

1
Take one term at a time
Integrate each term of the polynomial separately.
2
Raise the power
Add $1$ to the power of $x$ in the term.
3
Divide by the new power
Divide the coefficient by the new power.
4
Handle the constant term
A constant $k$ integrates to $kx$ .
5
Add c
For an indefinite integral, add the constant of integration $c$ .
6
Find c if a point is given
Substitute the given point to solve for $c$ , then write the final equation.

Worked example

Q1[5 marks]

The gradient function of a curve is $\frac{dy}{dx}=6x^{2}-4x+3$ , and the curve passes through the point $(1,4)$ . Find the equation of the curve.

Show worked solution

Integrate the gradient function term by term to recover $y$ .

For $6x^{2}$ : raise the power to $3$ and divide by $3$ : $\frac{6x^{3}}{3}=2x^{3}$ .

For $-4x$ : raise the power to $2$ and divide by $2$ : $\frac{-4x^{2}}{2}=-2x^{2}$ .

For the constant $3$ : it integrates to $3x$ .

y=\int(6x^{2}-4x+3)\,dx=2x^{3}-2x^{2}+3x+c

Use the point $(1,4)$ to find $c$ . Substitute $x=1$ , $y=4$ :

4=2(1)^{3}-2(1)^{2}+3(1)+c=2-2+3+c=3+c

So $c=1$ . The equation of the curve is:

y=2x^{3}-2x^{2}+3x+1

Check: at $x=1$ , $y=2-2+3+1=4$ , which matches the given point.

Common mistakes to avoid

Forgetting the constant of integration $c$ on an indefinite integral, a very common lost mark.
Confusing integration with differentiation: here you raise the power and divide, you do not lower it and multiply.
Dividing by the old power instead of the new power, e.g. writing $\frac{6x^{3}}{2}$ instead of $\frac{6x^{3}}{3}$ .
Integrating a constant term to another constant instead of to $kx$ .
Leaving $+c$ in the final answer when a point was given, instead of solving for its value.

How one-to-one teaching helps

The step students most often drop is the constant of integration, and then the missed method of using a point to find it. In a one-to-one lesson our teachers make writing $+c$ a reflex, then walk you through substituting the given point so that finding $c$ feels routine rather than an afterthought.

We also pin down the direction of the rule, raise the power, divide by the new power, because reversing the differentiation rule by mistake is the classic slip. Since the paper awards method marks, a clear integration line still scores even if the arithmetic wobbles.

Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English. To drill integration cleanly, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.

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Frequently asked questions

Why do I add

+c

when I integrate?

Differentiating a constant gives zero, so when you reverse the process you cannot know what constant was there. The $+c$ , called the constant of integration, stands for every possible constant until extra information, such as a point on the curve, fixes its value.

How is integrating a polynomial different from differentiating one?

They are opposite processes. To differentiate a term you lower the power by one and multiply by the old power.

To integrate you raise the power by one and divide by the new power, then add $c$ . Getting the direction right is the key to avoiding errors.

How do I find the value of

c

If the question gives a point that the curve passes through, substitute its $x$ - and $y$ -values into your integrated expression and solve the resulting equation for $c$ . Then write the full equation with that value in place.

What about a definite integral with limits?

Integrate exactly the same way, but do not add $c$ . Instead substitute the upper limit and the lower limit into the integrated expression and subtract: $F(b)-F(a)$ .

The constant would cancel, which is why it is omitted for definite integrals.

Source:SRC-DSKP-EN

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

How to integrate a polynomial