Math ↠ Definite integral ↠ Rules

Rules for definite integrals

When solving definite integrals, there are some rules that you should be able to apply.

  1. Same lower and upper limit

    $\int_a^a f(x) \, \mathrm{d}x=0$

  2. Reversing the limits

    $\int_a^b f(x) \, \mathrm{d}x$ $=-\int_b^a f(x) \, \mathrm{d}x$

  3. Additive interval

    $\int_a^b f(x) \, \mathrm{d}x + \int_b^c f(x) \, \mathrm{d}x$ $=\int_a^c f(x) \, \mathrm{d}x$

The constant factor rule and sum rule for indefinite integrals are also valid for the definite integrals.

  1. Constant factor rule

    $\int_a^b k\cdot f(x) \, \mathrm{d}x$ $= k\cdot \int_a^b f(x) \, \mathrm{d}x$

  2. Sum rule

    $\int_a^b (f(x)+g(x)) \, \mathrm{d}x$ $= \int_a^b f(x) \, \mathrm{d}x + \int_a^b g(x) \, \mathrm{d}x$