Properties of 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 trivial 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}



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