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

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