1 Handy inequalities
Young’s inequality for products:
If
My favourite generalisation: If
2 exp, sinh, cos etc
Avoid having to integrate by parts twice
Suppose
and are functions that are each proportional to their second derivative. These include exponential, circular, and hyperbolic functions. Then the integral of can be computed in closed form with a moderate amount of work. There’s a formula that can compute all these related integrals in one fell swoop. (Pease 1959) Suppose and for constants and . … Then
3 Maclaurin integration
consider the integral
The most common approach to evaluating this integral is to expand it as a power series and integrate term-by-term, which yields as the antiderivative, with as the constant of integration. Maclaurin Integration is an alternative solution by series (although not a power series, since it involves the function itself in the solution) that eliminates the nuisance of calculation completely.
…the formula is simply:
where is the constant of integration.… This formula is valid only if: 1.
is defined on the domain , 2. is continuous on , and 3. has derivatives of all orders on .