Example 7.2.6(a): Applying Integration by Parts
Find x ex dxWe need to identify two functions such that
In our case we define g'(x) = ex and f(x) = x. Then we need to find G(x) = f(x) g(x), which in this case is G(x) = x ex.
- we know the antiderivative of the first function - that function will be g'.
- the derivative of the second function is easier than the original function - that function will be f.
Integration by parts now gives the answer:
x ex dx = G(b) - G(a) - ex dx =where G(x) = x ex.
= G(b) - G(a) - [ exp(b) - exp(a) ]