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 formula: u d v = u v - v d u 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 above.
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 approach.
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 multiplication
  • T - Trigonometric functions
  • E - Exponential functions

In this case, we evaluate:
x csc 2 x d x
Integration by Parts.
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