In some cases, the function neither resembles a form
that matches any of the general integration patterns, and use of
the substitution technique doesn't work. In these cases, an
alternative approach is called INTEGRATION BY PARTS.
The use of integration by parts is based on the followng
To integrate by parts, we assign part of the function to a
variable 'u', and another part of the function to 'dv'. We then
find the derivative of 'u' ('du'), and the antiderivative of 'dv'
(to find 'v'), and place the results into the formula shown
The key is to select values for 'u' and 'dv' that result in
simpler integration steps as all the values are replaced into the
original function. Many instructors recommend using the ILATE
Each of the first letters describe a type of term, the term
that matches the sequence of ILATE is the term to be used as 'u':
- I - Inverse trig functions such as acrsin.
- L - Logarithms
- A - Algebraic terms such as terms involving simple
- T - Trigonometric functions
- E - Exponential functions
In this case, we evaluate: