Cho ∫ − 3 0 f ( x ) d x = − 4 và ∫ 0 5 f ( x ) d x = − 3 . ∫ − 3 5 f ( x ) d x

{\displaystyle \underset{-3}{\overset{5}{\int }}f\left(x\right)\text{d}x}={\displaystyle \underset{-3}{\overset{0}{\int }}f\left(x\right)\text{d}x+}{\displaystyle \underset{0}{\overset{5}{\int }}f\left(x\right)\text{d}x}.

{\displaystyle \underset{-3}{\overset{5}{\int }}\left(2f\left(x\right)+3\right)\text{d}x}=10.

${\displaystyle \underset{-3}{\overset{5}{\int }}\left(2f\left(x\right)\text{+}\text{3}\text{g}\left(x\right)\right)\text{d}x}=-26$ với ${\displaystyle \underset{-3}{\overset{5}{\int }}2g\left(x\right)\text{d}x}=-4$.

Nếu $f\left(x\right)=4x+3$ thì {\displaystyle \underset{0}{\overset{a}{\int }}f\left(x\right)\text{d}x}=5\iff a\in \left(-3;1\right).