Integrating by parts is a method used in calculus to find the integral of a product of two functions. It is particularly useful when the integral involves a product of a polynomial and an exponential function.
The method of integrating by parts involves using the product rule of differentiation in reverse. It allows us to rewrite the integral as a simpler one, which can be easily evaluated.
When using the integrating by parts method, it is important to choose a function to differentiate and a function to integrate that will lead to a simpler integral. In this case, choosing the polynomial function x^4 as the one to differentiate and the exponential function e^x as the one to integrate leads to the simplest integral.
Yes, other numbers can be used for \int {x^4 e^x dx}. However, choosing a number other than 4 may result in a more complicated integral that is more difficult to evaluate.
The integrating by parts method can still be used for integrals involving different combinations of functions. It is important to carefully choose which function to differentiate and which function to integrate in order to simplify the integral as much as possible.