Basic Rules of Definite Integrals
Basic Rules of Definite Integrals
IB AA SL: core and optional content
Core: definite-integral properties when the integrals exist. Optional: formal one-sided improper-integral analysis.
\(\displaystyle\displaystyle \displaystyle\int_{b}^{a} \mathrm{f}(x) \, \mathrm{d}x = \mathrm{F}(a) - \mathrm{F}(b)\) \(\displaystyle\displaystyle \displaystyle\int_{b}^{a} \mathrm{f}(x) \, \mathrm{d}x = -\displaystyle\displaystyle \displaystyle\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x\) \(\displaystyle\displaystyle \displaystyle\int_{a}^{c} \mathrm{f}(x) \, \mathrm{d}x = \displaystyle\displaystyle \displaystyle\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x + \displaystyle\displaystyle \displaystyle\int_{b}^{c} \mathrm{f}(x) \, \mathrm{d}x\) |
These rules require the relevant integrals to exist. For a continuous integrand on the closed interval, an antiderivative gives \(\int_a^b f(x)\,\mathrm dx=F(b)-F(a)\). If the integrand is unbounded or undefined inside the interval or at an endpoint, first analyse the appropriate one-sided improper integrals; substituting endpoints alone is not sufficient.
Example
\(\displaystyle\int_1^3(2x+1)\,\mathrm dx=\left[x^2+x\right]_1^3=12-2=10\), so \(\displaystyle\int_3^1(2x+1)\,\mathrm dx=-10\).
\(\displaystyle\int_1^2(2x+1)\,\mathrm dx+\int_2^3(2x+1)\,\mathrm dx=4+6=10\), the same as over \([1,3]\).
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