## Formula deduction for piecewise integration

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

Integrating both sides, we get

or

or

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

(1) |

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):

(2) |

**Example**

Calculate

** 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