# Indefinite integration of polynomials

> IB Mathematics: Applications and Interpretation SL · IB AI SL Unit 5: Calculus
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u5-indefinite-integration-of-polynomials/

This sub-topic covers finding antiderivatives (indefinite integrals) of polynomial functions using the power rule for integration. You will also learn how to calculate the constant of integration when given an initial condition.

**Prerequisites:** [Power rule for differentiation of polynomials](https://www.owlsprep.com/study/ib-math-ai-sl-u4-differentiation-of-polynomials/); Basic polynomial algebraic operations

## Learning objectives

- Recall the definition of an antiderivative and indefinite integral
- Apply the power rule to find indefinite integrals of polynomials
- Explain the purpose of the constant of integration
- Solve for the constant of integration given an initial condition

## What is Indefinite Integration?

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

*Notation:* \int f(x)dx

*Example:* One antiderivative of $2x$ 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 $F(x) = 2x^3 + 4x - 5$ is an antiderivative of $f(x) = 6x^2 + 4$

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.

## The Power Rule for Integration

**Power Rule for Indefinite Integration** — 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.

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

*Example:* $\int 3x^2 dx = 3 \cdot \frac{x^3}{3} + C = x^3 + C$

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

> **tip**
>
> You can always check your integration result by differentiating it — you should get back the original function if you did it correctly.

**Worked example:** Find the indefinite integral of $f(x) = 3x^2 + 4x - 5$

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. $$\int (3x^2 + 4x -5) dx = x^3 + 2x^2 - 5x + C$$

## Finding the Constant of Integration

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 $\frac{dy}{dx} = 2x + 3$, 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. $$y = x^2 + 3x + 1$$

**Check your understanding**

Test your understanding

1. If $y = \int f(x) dx = 2x^2 + 3x + C$, and $y = 10$ when $x = 2$, what is $C$?

   - 10
   - -4
   - 0
   - 4

   *Why:* 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 $C$.

## Common pitfalls

- **Wrong:** Forgetting to add the constant of integration $C$ to the final result
  - Why it fails: Examiners almost always deduct 1 mark for omitting $C$, even if all other working is correct
  - Correct: Write $+C$ at the end of every indefinite integral, before solving for a specific $C$ from an initial condition
- **Wrong:** Applying the power rule incorrectly, multiplying by $n+1$ instead of dividing
  - Why it fails: Confusing the integration power rule with differentiation, where you multiply by the original power
  - Correct: Remember the rule: add 1 to the power, divide by the new power
- **Wrong:** Integrating a constant term $k$ as just $C$, instead of $kx + C$
  - Why it fails: Confusing the constant being integrated with the arbitrary constant of integration
  - Correct: Treat $k$ as $kx^0$, so it integrates to $kx$, then add $C$ to the full result
- **Wrong:** Substituting the initial condition into the original derivative instead of the antiderivative
  - Why it fails: Mixing up what each function represents; the initial condition is a point on the antiderivative curve
  - Correct: Always substitute the given $(x,y)$ values into the integrated function to solve for $C$

## Cheatsheet

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

## 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.

- [Definite integrals and area under a curve](https://www.owlsprep.com/study/ib-math-ai-sl-u5-definite-integrals-and-area-under/)
- [Kinematic applications of calculus](https://www.owlsprep.com/study/ib-math-ai-sl-u5-kinematic-applications-of-calculus/)

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