| S.No | Property |
| 1. | $$\int\limits_a^b {f\left( x \right)dx} = \int\limits_a^b {f\left( t \right)dt} $$ |
| 2. | $$\int\limits_a^b {f\left( x \right)dx} = - \int\limits_b^a {f\left( x \right)dx} $$ |
| 3. | $$\int\limits_a^b {f\left( x \right)dx} = \int\limits_a^c {f\left( x \right)dx} + \int\limits_c^b {f\left( x \right)dx} $$ |
| 4. | $$\int\limits_a^b {f\left( x \right)dx} = \int\limits_a^b {f\left( {a + b - x} \right)dx} $$ |
| 5. | $$\int\limits_0^a {f\left( x \right)dx} = \int\limits_0^a {f\left( {a - x} \right)dx} $$ |
| 6. | $$\int\limits_0^{2a} {f\left( x \right)dx} = \int\limits_0^a {f\left( x \right)dx} + \int\limits_0^a {f\left( {2a - x} \right)dx} $$ |
| 7. | $$\int\limits_0^{2a} {f\left( x \right)dx} = 2\int\limits_0^a {f\left( x \right)dx} $$ if $f\left( {2a - x} \right) = f\left( x \right)$ |
| $$\int\limits_0^{2a} {f\left( x \right)dx} = 0$$ if $f\left( {2a - x} \right) = - f\left( x \right)$ |
| 8. | $$\int\limits_{ - a}^a {f\left( x \right)dx} = 2\int\limits_0^a {f\left( x \right)dx} $$ if $f$ is an even function i.e. $f\left( { - x} \right) = f\left( x \right)$ |
| $$\int\limits_{ - a}^a {f\left( x \right)dx} = 0$$ if $f$ is an odd function i.e. $f\left( { - x} \right) = - f\left( x \right)$ |
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