∫ udv uv vdu This method is very useful when the integrand is a product of two different types of functions or a function which is not directly integrable. The success of this method depends on the proper choice of u . So we can choose the function u by using the following guidelines. (i) If the integrand contains only a function which is directly not integrable, then take this as u .
(ii) If the integrand contains both directly integrable and non integrable functions, then take non integrable function as u . (iii) If the integrand contains both the functions are integrable and one of them is of the form x n , is a positive integer, then take this x n as u . (iv) for all other cases, the choice of u is ours (Or) we can also choose u as the function which comes first in the word “I L A T E” Where, I stands for the inverse trigonometric function L stands for the logarithmic function A stands for the algebraic function T stands for the trigonometric function E stands for the exponential function and take the remaining part of the function and dx as dv . ( ) If u and v are functions of x , then udv uv u v u v u v −′ + ′′ −′′′ ...
where ′ ′′ ′′′ u u u , . . . are the successive derivatives of u and v v v , .
. . are the repeated integrals of v . The above mentioned formula is well known as Bernoulli’s formula.
Bernoulli’s formula is applied when u x n where n is a positive integer. Note Example . Evaluate xe dx