Integration by parts:

If u and v be 2 functions of x, then integral of product of these 2 functions can be given by: 

Note:   In applying above rule care has to be taken in the selection of the first function(u) and the second function (v). Normally we use the following methods:

If in the product of the two functions, one of these functions is not directly integrable
(for instance lnx, sin^-1x, cos^-1x, tan^-1x etc.) then we take it as the 1^st function and the remaining function is taken as the second function. For example In the integration of òx tan^-1x dx, tan^-1x is taken as the 1^st function and x as the 2^nd function.

If there exists no other function, then the unity is taken as the 2^nd function for example In the integration of òtan^-1x dx , tan^-1x is taken as the 1^st function and 1 as the 2^nd function.

If both of the function are directly integrable then the 1^st function is chosen in such a manner that the derivative of function thus obtained under integral sign is integrable easily. Usually we use the following preference order for the 1st function

(Inverse, Logarithmic, Algebraic, Trigonometric, Exponent)

In the above stated order, function on the left is always chosen as 1^st function. This rule is known as ILATE for example In the integration of ∫x sinx xdx, x is taken as the 1^st function and sinx is taken as the 2^nd function.

Example:  Evaluate .

Solution:      

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