In details

Piecewise integration

Formula deduction for piecewise integration

If f and g are differentiable functions, so by the product differentiation rule,

Integrating both sides, we get



Since the right integral will produce another integration constant, there is no need to keep C in this last equation; so we get


which is called piecewise integration formula. Using this formula, we can sometimes make an integration problem simpler.

In practice, it is usual to rewrite (1)

u = f (x), du = f '(x) dx


This gives rise to the following alternative form for (1):




Solution. To apply (2), we need to write the integral in the form

One way to do this is to put

for what,

Thus, from (2)

Next: Piecewise Integration for Defined Integrals