# Linearity of Integrals: Sum, Difference and Constant Multiple Rules

> The integral linearity rules: integrate sums term by term and pull constants out. Formulas, examples and the mistakes to avoid with products and quotients.

Source: https://practicecalculus.com/rules/integral-linearity-rules
Updated: 2026-11-03

$$ \int \left[a\,f(x) + b\,g(x)\right]dx = a\int f(x)\,dx + b\int g(x)\,dx $$

## At a glance

- **Sum rule:** $\int (f + g)\,dx = \int f\,dx + \int g\,dx$
- **Difference rule:** $\int (f - g)\,dx = \int f\,dx - \int g\,dx$
- **Constant multiple rule:** $\int k\,f(x)\,dx = k\int f(x)\,dx$ for any constant $k$
- **Does not apply to:** Products and quotients: $\int f g\,dx \ne \int f\,dx \cdot \int g\,dx$
- **Holds for:** Both indefinite and definite integrals

## Key takeaways

- You can integrate a sum or difference one term at a time.
- Constant factors can be moved outside the integral sign.
- There is no product or quotient rule for integrals: expand, split, or use a technique such as u-substitution.

The **linearity rules** say that integration respects addition and scaling. You can split a sum into separate integrals and move constant factors outside the integral sign. Together with the [reverse power rule](/rules/reverse-power-rule), they let you integrate any polynomial in one line.

## The linearity rules

For integrable functions $f$ and $g$ and constants $a$ and $b$:

$$\int \left[a\,f(x) + b\,g(x)\right]dx = a\int f(x)\,dx + b\int g(x)\,dx$$

This single statement contains three rules:

- **Sum rule:** $\int (f + g)\,dx = \int f\,dx + \int g\,dx$
- **Difference rule:** $\int (f - g)\,dx = \int f\,dx - \int g\,dx$
- **Constant multiple rule:** $\int k\,f(x)\,dx = k\int f(x)\,dx$

The same rules hold for definite integrals over the same interval:

$$\int_a^b \left[f(x) + g(x)\right]dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx$$

## Why linearity works

Differentiation is linear: $\frac{d}{dx}[aF + bG] = aF' + bG'$. If $F' = f$ and $G' = g$, then $aF + bG$ is an antiderivative of $af + bg$, which is exactly the rule. For definite integrals the reason is even more visual: a Riemann sum of $f + g$ is the sum of the Riemann sums of $f$ and $g$, and stretching a graph by $k$ stretches every rectangle, and so the area, by $k$.

```graph
title: 'Area under y = 2x² compared with the area under y = x² on [0, 2]'
caption: 'Doubling the function doubles the area: $\int_0^2 2x^2\,dx = 2 \cdot \frac{8}{3} = \frac{16}{3}$.'
x: [-0.5, 2.5]
y: [-1, 9]
functions:
  - expr: 2x^2
    label: y = 2x²
  - expr: x^2
    label: y = x²
    dashed: true
shade: {expr: 2x^2, from: 0, to: 2}
```

## Worked examples

**Example 1: a polynomial.** Split, pull out constants, then apply the reverse power rule to each piece:

$$\int \left(8x^3 - 6x + 7\right)dx = 8\int x^3\,dx - 6\int x\,dx + \int 7\,dx = 2x^4 - 3x^2 + 7x + C$$

**Example 2: mixed function types.** Linearity lets you combine rules from different families:

$$\int \left(3\cos x - \frac{2}{x} + 4e^x\right)dx = 3\sin x - 2\ln\lvert x\rvert + 4e^x + C$$

**Example 3: split a quotient first.** There is no quotient rule for integrals, but you can divide each term by the denominator:

$$\int \frac{x^2 + 3x - 1}{x}\,dx = \int \left(x + 3 - \frac{1}{x}\right)dx = \frac{x^2}{2} + 3x - \ln\lvert x\rvert + C$$

**Example 4: a definite integral.** Using known values $\int_0^1 f(x)\,dx = 4$ and $\int_0^1 g(x)\,dx = -1$:

$$\int_0^1 \left[2f(x) - 5g(x)\right]dx = 2(4) - 5(-1) = 13$$

## What linearity does not do

Linearity covers sums and constant multiples only. These are all **false** in general:

| Tempting but wrong | Why |
| --- | --- |
| $\int f g\,dx = \int f\,dx \cdot \int g\,dx$ | No product rule for integrals |
| $\int \frac{f}{g}\,dx = \frac{\int f\,dx}{\int g\,dx}$ | No quotient rule for integrals |
| $\int x\,f(x)\,dx = x\int f(x)\,dx$ | $x$ is not a constant |
| $\int \frac{1}{f(x)}\,dx = \frac{1}{\int f(x)\,dx}$ | Reciprocals do not pass through |

When the integrand is a product, try expanding it, then reach for [u-substitution](/rules/u-substitution-rule) or [integration by parts](/rules/integration-by-parts-rule).

## Common mistakes

1. **Moving the variable outside.** Only constants come out. In $\int 3x\sin x\,dx$ you may pull out the $3$, not the $x$.
2. **Integrating a product factor by factor.** $\int x(x^2+1)\,dx$ must be expanded to $\int (x^3 + x)\,dx = \frac{x^4}{4} + \frac{x^2}{2} + C$.
3. **Splitting a denominator.** $\frac{1}{x + 1} \ne \frac{1}{x} + 1$. You can split a fraction across terms in the numerator, never the denominator.
4. **Losing a sign in a difference.** Distribute the minus sign to every term of the second integral.

The derivative counterparts are the [constant multiple rule](/rules/constant-multiple-rule) and the [sum and difference rule](/rules/sum-and-difference-rule). For more practice combining these rules, see the guide to [antiderivatives](/blog/antiderivatives).

## Frequently asked questions

### What does linearity of the integral mean?

It means the integral of $af + bg$ equals $a$ times the integral of $f$ plus $b$ times the integral of $g$, for constants $a$ and $b$. In practice you split sums apart and pull constants out.

### Can you pull a variable out of an integral?

No. Only constants, meaning quantities that do not depend on the variable of integration, can move outside. $\int x\cos x\,dx$ is not $x\int\cos x\,dx$.

### Is the integral of a product the product of the integrals?

No. For example $\int x \cdot x\,dx = \frac{x^3}{3} + C$, but $\int x\,dx \cdot \int x\,dx = \frac{x^4}{4}$ (up to constants). Products need algebra or integration by parts.

### Do I need a separate +C for each term?

No. The separate constants add up to one arbitrary constant, so you write a single $+C$ at the end.