7.2. Integration Techniques

Example 7.2.6(a): Applying Integration by Parts

Find x ex dx
We need to identify two functions such that
  • 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.
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.

Integration by parts now gives the answer:

x ex dx = G(b) - G(a) - ex dx =
      = G(b) - G(a) - [ exp(b) - exp(a) ]
where G(x) = x ex.
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