Indefinite integration of polynomials

IB Mathematics: Applications and Interpretation SL· 35 min read

1. What is Indefinite Integration?★☆☆☆☆⏱ 10 min

📘 Definition

Indefinite Integral

\int f(x)dx

The collection of all possible antiderivatives of a function $f (x)$ . The derivative of any antiderivative in the collection equals the original function $f (x)$ .

Example:

One antiderivative of $2 x$ is $x^{2}$ , another is $x^{2} + 3$ , so the indefinite integral is $x^{2} + C$

Integration reverses the process of differentiation. If you know the derivative of a function, integration lets you recover the original function, up to an unknown constant. Because the derivative of any constant is zero, adding a constant to a function does not change its derivative. This is why all indefinite integrals include an arbitrary constant of integration $C$ .

📐 Worked Example

Confirm that $F (x) = 2 x^{3} + 4 x - 5$ is an antiderivative of $f (x) = 6 x^{2} + 4$

1
Differentiate $F (x)$ term by term using the power rule for differentiation
2
$F'(x) = \frac{d}{dx}\left(2x^3\right) + \frac{d}{dx}\left(4x\right) - \frac{d}{dx}\left(5\right) = 6x^2 + 4 - 0$
3
Since $F^{'} (x) = f (x)$ , $F (x)$ is confirmed as an antiderivative of $f (x)$

Exam tip:

Always add $+ C$ to your final indefinite integral result — examiners regularly deduct marks for omitting it.

2. The Power Rule for Integration★★☆☆☆⏱ 15 min

📘 Definition

Power Rule for Indefinite Integration

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

For any real number $n$ not equal to $- 1$ , the indefinite integral of $x^{n}$ is found by adding 1 to the exponent, then dividing by the new exponent, plus the constant of integration.

Example:

$\int 3 x^{2} d x = 3 \cdot \frac{x ^{3}}{3} + C = x^{3} + C$

To integrate a full polynomial, integrate each term separately, just like you differentiate term-by-term. Integration is linear, meaning $\int (a f (x) + b g (x)) d x = a \int f (x) d x + b \int g (x) d x$ .

📐 Worked Example

Find the indefinite integral of $f (x) = 3 x^{2} + 4 x - 5$

1
Integrate each term separately using the power rule
2
$\int 3x^2 dx = 3 \cdot \frac{x^{2+1}}{2+1} = x^3$
3
$\int 4x dx = 4 \cdot \frac{x^{1+1}}{1+1} = 2x^2$
4
$\int -5 dx = \int -5x^0 dx = -5 \cdot \frac{x^{0+1}}{0+1} = -5x$
5
Combine results and add the constant of integration $C$
6
$\int (3x^2 + 4x -5) dx = x^3 + 2x^2 - 5x + C$

3. Finding the Constant of Integration★★★☆☆⏱ 15 min

If you are given an initial condition (a point that the antiderivative passes through), you can solve for the specific value of $C$ , which gives a unique antiderivative instead of a family of curves. This is a very common exam question.

📐 Worked Example

Given $\frac{d y}{d x} = 2 x + 3$ , and $y = 5$ when $x = 1$ , find $y$ in terms of $x$

1
First find the general indefinite integral to get the general form of $y$
2
$y = \int (2x + 3) dx = 2 \cdot \frac{x^2}{2} + 3x + C = x^2 + 3x + C$
3
Substitute the given values $x = 1, y = 5$ into the equation to solve for $C$
4
$5 = (1)^2 + 3(1) + C \rightarrow 5 = 1 + 3 + C$
5
Solve for $C$ : $C = 5 - 4 = 1$
6
Write the final unique antiderivative
7
$y = x^2 + 3x + 1$

✓ Quick check

Test your understanding

If $y = \int f (x) d x = 2 x^{2} + 3 x + C$ , and $y = 10$ when $x = 2$ , what is $C$ ?
- 10
- -4
- 0
- 4
Reveal answer
-4 —
Substitute $x = 2, y = 10$ : $10 = 2 (2)^{2} + 3 (2) + C = 8 + 6 + C$ , so $C = 10 - 14 = - 4$

Exam tip:

Always substitute the initial condition into the antiderivative (integrated function), not the original derivative, to find $C$ .

4. Common Pitfalls

Wrong move:

Forgetting to add the constant of integration $C$ to the final result

Why:

Examiners almost always deduct 1 mark for omitting $C$ , even if all other working is correct

Correct move:

Write $+ C$ at the end of every indefinite integral, before solving for a specific $C$ from an initial condition

Wrong move:

Applying the power rule incorrectly, multiplying by $n + 1$ instead of dividing

Why:

Confusing the integration power rule with differentiation, where you multiply by the original power

Correct move:

Remember the rule: add 1 to the power, divide by the new power

Wrong move:

Integrating a constant term $k$ as just $C$ , instead of $k x + C$

Why:

Confusing the constant being integrated with the arbitrary constant of integration

Correct move:

Treat $k$ as $k x^{0}$ , so it integrates to $k x$ , then add $C$ to the full result

Wrong move:

Substituting the initial condition into the original derivative instead of the antiderivative

Why:

Mixing up what each function represents; the initial condition is a point on the antiderivative curve

Correct move:

Always substitute the given $(x, y)$ values into the integrated function to solve for $C$

5. Quick Reference Cheatsheet

Rule	Formula	Key Note
Power Rule ( $n \neq = - 1$ )	$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$	Add 1 to power, divide by new power
Constant Multiple	$\int k f (x) d x = k \int f (x) d x$	Factor out constants before integrating
Term-by-Term Integration	$\int (f + g) d x = \int f d x + \int g d x$	Integrate each polynomial term separately
Find $C$	Substitute $(x, y)$ into $F (x) = \int f (x) d x + C$	Initial condition is always on $F (x)$ , not $f (x)$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find C given initial condition
2022 · 1
Integrate cubic polynomial
2023 · 2
Integrate and find specific antiderivative

What's Next

Mastering indefinite integration of polynomials is the foundation for all further integration topics in IB AI SL. Next, you will move on to definite integration, which lets you calculate the area under a curve — one of the most common applied uses of calculus. This skill is also essential for solving differential equations, which are used to model real-world phenomena like population growth, cooling, and radioactive decay, a key topic for your IA and final exam.