If f(x) and g(x) are two integrable functions, then
∫ f(x).g(x) dx = f(x) ∫ g(x) dx – ∫ f(x) [∫ g(x) dx] dx.
(i) If both the functions are directly integrable, then the first function will be taken in the following order:
Inverse functions, logarithmic functions, algebraic functions, trigonometric functions and exponential functions.
Mnemonic to remember the order is ILATE.
Ex: ∫ x ex dx; ∫ sin x.ex dx; ∫ x.sin x dx
(ii) In the product of two functions, if one of the function is not directly integrable (i.e., log |x|, sin–1 x. cos–1 x;
tan–1 x, ... etc), we take it as the first function and the other function is taken as the second function.
Ex: ∫ x.sin–1 x dx; ∫ ex log x dx
(iii) If there is no other function, then unity is taken as the second function.
Ex: ∫ log x dx; ∫ sin–1 x dx