Chasle's relation for definite integrals

Theorem. (Chasle's relation)

Let \(\mathrm{f}\) be a function defined over the interval \([a,b]\), where \(c \in [a,b]\). Then,
\[ \displaystyle \displaystyle\int_a^c \mathrm{f}(x) \, \mathrm{d}x + \displaystyle \displaystyle\int_c^b \mathrm{f}(x) \, \mathrm{d}x = \displaystyle \displaystyle\int_a^b \mathrm{f}(x) \, \mathrm{d}x. \]

In particular,
\[ \displaystyle \displaystyle\int_a^b \mathrm{f}(x) \, \mathrm{d}x = - \displaystyle \displaystyle\int_b^a \mathrm{f}(x) \, \mathrm{d}x. \]