14.3 Summary
In this chapter, we have learned about integration, one of the technically most challenging subjects so far. Intuitively, the integral of a function describes the signed area under its graph, but mathematically, it is given by the limit
where a = x0/span>x1/span>…/span>xn = b is a partition of the interval [a,b]. Of course, we don’t often calculate integrals by the definition; we have the Newton-Leibniz formula for that:
where F is the so-called antiderivative, satisfying F′(x) = f(x). This is why integration is thought of as the inverse of differentiation.
As one of my professors used to say, symbolic differentiation is easy, numeric differentiation is hard. It’s the opposite for integrals: symbolic integration is hard, and numeric integration is easy. We’ve learned a couple of tricks to pin down the symbolic part, namely the integration by parts formula
![∫ ∫ b ′ x=b b ′ a f(x)g(x)dx = [f(x)g(x)]x=a − a f(x)g (x)dx](https://static.packt-cdn.com/products/9781837027873/graphics/media/file1420.png)
and the integration by substitution formula
When symbolic integration...
