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