Choose substitution, inverse forms, partial fractions or integration by parts. Set up geometric integrals with the correct element of length. Express a rectangular sum using standard sums and pass to its limit.
Notes and key methods
A substitution must transform the whole integrand, differential and limits. Recognise \(\int x/(x^2+a^2)\,dx=\frac12\ln(x^2+a^2)+C\), and \(\int dx/\sqrt{x^2+a^2}=\operatorname{arsinh}(x/a)+C\) for a>0.
Use integration by parts \(\int u\,dv=uv-\int v\,du\) when differentiating one factor simplifies it. A proper rational function may instead be decomposed into partial fractions. Check an antiderivative by differentiation.
For a smooth curve y=f(x), arc length is \(L=\int_a^b\sqrt{1+(dy/dx)^2}\,dx\). For revolution about the x-axis with y≥0, curved surface area is \(S=2\pi\int_a^b y\sqrt{1+(dy/dx)^2}\,dx\).
Volume uses \(\pi\int y^2dx\), a different geometric quantity. When working parametrically, replace the arc element by √[(dx/dt)²+(dy/dt)²]dt.
With n equal subintervals of width h=(b−a)/n, a right-endpoint sum is \(h\sum_{r=1}^n f(a+rh)\). For a continuous function this tends to the definite integral as n increases.
If f is increasing and nonnegative, the left sum is a lower bound and the right sum is an upper bound. Derive the finite expression before taking a limit, and keep the factor h outside the sum.
Check the conditions
Transform definite limits when changing variables.
Retain absolute values in logarithms where the domain needs them.
The curve is smooth on the stated interval.
Surface area uses a nonnegative radius.
The partition and endpoint convention are stated.
Bounds from monotonicity require the appropriate increasing or decreasing behaviour.
Three worked examples
Example 1 · Advanced integration techniques
Evaluate the following independent definite integrals on [0,1]. Show a suitable method.