A definite integral carries limits and evaluates to a number. If \(\displaystyle\int f(x)\,\mathrm{d}x=F(x)+c\), then \(\displaystyle\int_a^b f(x)\,\mathrm{d}x=\big[F(x)\big]_a^b=F(b)-F(a)\).
Evaluating a definite integral
| Step | Working |
|---|---|
| Integrate; write the result in square brackets with the limits | \(\displaystyle\int_1^3(2x+1)\,\mathrm{d}x=\big[x^2+x\big]_1^3\) |
| Substitute the upper limit, then subtract the lower | \(=(3^2+3)-(1^2+1)\) \(=12-2\) \(=10\) |
No \(+c\) is needed for a definite integral: it would appear in both brackets and cancel in the subtraction.
Properties of definite integrals
| Property | Statement | Example, using \(\displaystyle\int_0^4 f(x)\,\mathrm{d}x=7\) |
|---|---|---|
| Zero-length interval | \(\displaystyle\int_a^a f(x)\,\mathrm{d}x=0\) | \(\displaystyle\int_4^4 f(x)\,\mathrm{d}x=0\) |
| Reverse the limits | \(\displaystyle\int_a^b f(x)\,\mathrm{d}x=-\displaystyle\int_b^a f(x)\,\mathrm{d}x\) | \(\displaystyle\int_4^0 f(x)\,\mathrm{d}x=-7\) |
| Split the interval | \(\displaystyle\int_a^c f(x)\,\mathrm{d}x=\displaystyle\int_a^b f(x)\,\mathrm{d}x+\displaystyle\int_b^c f(x)\,\mathrm{d}x\) | \(\displaystyle\int_0^2 f(x)\,\mathrm{d}x+\displaystyle\int_2^4 f(x)\,\mathrm{d}x=7\) |
The constant-multiple and sum rules also hold between limits: \(\displaystyle\int_a^b k\,f(x)\,\mathrm{d}x=k\displaystyle\int_a^b f(x)\,\mathrm{d}x\) and \(\displaystyle\int_a^b[f(x)\pm g(x)]\,\mathrm{d}x=\displaystyle\int_a^b f(x)\,dx\pm\displaystyle\int_a^b g(x)\,\mathrm{d}x\).
Substitute the limits into the integrated expression, never into the original integrand, and keep the order: upper limit first, then subtract the lower.
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.
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