Numerical differentiation and integration

In this chapter we propose methods for the numerical approximation of derivatives and integrals of functions. Concerning integration, it is known that for a generic function it is not always possible to find a primitive in an explicit form. In other cases, it could be hard to evaluate a primitive as, for instance, in the example $$\frac{1}{\pi }\int\limits_{0}^{\pi } {\cos \left( {4x} \right)\cos \left( {3\sin \left( x \right)} \right)} dx = {{\left( {\frac{3}{2}} \right)}^{4}}\sum\limits_{{k = 0}}^{\infty } {\frac{{{{{\left( { - 9/4} \right)}}^{k}}}}{{k!\left( {k + 4} \right)!}},} $$ where the problem of computing an integral is transformed into the equally troublesome one of summing a series. It is also worth mentioning that sometimes the function that we want to integrate or differentiate could only be known on a set of nodes (for instance, when the latter represent the results of an experimental measurement), exactly as happens in the case of the approximation of functions, which was discussed in chapter 3.