Looking at the two functions involved in this example, we see that the function x is easy to integrate or differentiate. The real problem is how to handle . First it is not clear how to integrate this one but its derivative is the rational function . Therefore this suggests the following
The integration by parts formula gives
The new integral will be handled by using the technique of integrating rational functions. But we can also do the following (which comes up doing the same ideas used in partial fraction decomposition)
Using this we get
The main idea behind this example is valid for many other functions such as: , , etc.. In fact, this is how the integration by parts should be carried whenever the integral is given as a product of f(x) and one the previous inverse-functions, try to integrate f(x) and differentiate the inverse-function. The same remark holds for the function .
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Author: Mohamed Amine Khamsi