# Integration by Parts

Theorem: If

*u*and*v*are two functions of*x*, then =*u*â€“*dx.*

*Notes:*In applying the above rule, care has to be taken in the selection of the first function (

*u*) and the selection of second function (

*v*). Normally we use the following methods:

- If in the product of the two functions, one of the functions is not directly integrable (e.g., log |
*x*|, sin^{â€“1}*x*, cos^{â€“1}*x*, tan^{â€“1}*x*, â€¦, etc.), Then we take it as the first function and the remaining, function is taken as the second function. For example, in the integration of*x**dx*, tan^{â€“1}*x*is taken as the first function and*x*as the second function. - If there is no other function then unity is taken as the second function. For example, in the integration of , tan
^{â€“1}*x*is taken as the first function and 1 as the second function. - If both of the functions are directly integrable then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable.
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