# Properties of definite integral

## Rules

### Sum/Difference

$$
\int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx
$$

### Constant multiple

$$
\int_a^b k \cdot f(x) dx = k \int_a^b f(x) dx
$$

### Reverse interval

$$
\int_a^b f(x) dx = - \int_b^a f(x) dx
$$

### Zero-length interval

$$
\int_a^a f(x) dx = 0
$$

### Adding intervals

$$
\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx
$$

## Example

Given $$\int_{-3}^{1} g(x) dx = 6$$ and $$\int_{1}^{5} g(x) dx = 8$$

$$\int_{-3}^5 g(x) dx = 6 + 8 = 14$$

Given $$\int_{-1}^{3} f(x) dx = -2$$ and $$\int_{-1}^{3} g(x) dx = 5$$

$$\int_{-1}^{3} (3f(x) - 2g(x)) dx = -6 - 10 = -16$$

Given $$\int_{-5}^{1} g(x) dx = 3$$ and $$\int_{-3}^{1} g(x) dx = 7$$

$$\int_{-5}^{-3} g(x) dx = 3 - 7 = -4$$
