# Properties of the Definite Integral

From the definition of the definite integral $\int\limits_a^b {f\left( x \right)dx = F\left( b \right) - F\left( a \right)}$, we have the following results.

1. $\int\limits_a^b {f\left( x \right)dx = } - \int\limits_b^a {f\left( x \right)dx}$

2. $\int\limits_a^b {\left[ {f\left( x \right) + g\left( x \right)} \right]dx = } \int\limits_b^a {f\left( x \right)dx} + \int\limits_b^a {g\left( x \right)dx}$

3. $\int\limits_a^b {\left[ {f\left( x \right) - g\left( x \right)} \right]dx = } \int\limits_b^a {f\left( x \right)dx} - \int\limits_b^a {g\left( x \right)dx}$

4. $\int\limits_a^b {f\left( x \right)dx = } \int\limits_a^c {f\left( x \right)dx} + \int\limits_c^b {g\left( x \right)dx,\,\,\,a < c < b}$

5. $\int\limits_a^b {kf\left( x \right)dx = } k\int\limits_a^b {f\left( x \right)dx}$

6. If $f\left( x \right) \leqslant g\left( x \right)$ for $x \in \left[ {a,b} \right]$ $\Rightarrow \int\limits_a^b {f\left( x \right)dx \leqslant } \int\limits_a^b {g\left( x \right)dx}$