Definite Integrals and Changed Limits

For a continuous integrand, use an antiderivative and subtract the lower-end value from the upper-end value.

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

Under \(u=\mathrm{g}(x)\), replace \(x=a\) by \(u=\mathrm{g}(a)\) and \(x=b\) by \(u=\mathrm{g}(b)\). Keep the differential and both limits in the same variable.

\(\begin{aligned}u&=1+x^2,\qquad\mathrm{d}u=2x\,\mathrm{d}x\\\int_0^1\dfrac{2x}{1+x^2}\,\mathrm{d}x&=\int_1^2\dfrac{1}{u}\,\mathrm{d}u\\&=\ln2\end{aligned}\)

For a supplied substitution \(x=a\sin\theta\), choose the stated or appropriate interval for \(\theta\) before replacing \(\sqrt{a^2-x^2}\). In general, \(\sqrt{a^2\cos^2\theta}=a|\cos\theta|\) for \(a>0\).

A decreasing substitution reverses the numerical order of the bounds; its negative differential accounts for that. Keep exact bounds until the final calculation.

Some definite values require numerical integration on a graphic display calculator. Use this when permitted and suitable; retain exact working when an exact value is requested.