Theorem. (Chasle's relation)
Let \(\mathrm{f}\) be a function defined over the interval \([a,b]\), where \(c \in [a,b]\). Then, \[
\displaystyle\int\nolimits_a^c \mathrm{f}(x) \, \mathrm{d}x + \displaystyle\int\nolimits_c^b \mathrm{f}(x) \, \mathrm{d}x = 
\displaystyle\int\nolimits_a^b \mathrm{f}(x) \, \mathrm{d}x.
\]
In particular, \[
\displaystyle\int\nolimits_a^b \mathrm{f}(x) \, \mathrm{d}x = - \displaystyle\int\nolimits_b^a \mathrm{f}(x) \, \mathrm{d}x.
\]

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

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

Chasle's relation for definite integrals

Chasle's relation for definite integrals