Question

is there a fast way to integrate such integration without using integration by parts ? because its going to take too long..

for example integration for: x^4 e^x dx

Answer 1

\[\int\limits_{?}^{?} x^4 e^x dx\]

Answer 2

first see which one is reducing function. reducing function is the function which will ultimately become zero on successive differentiation.

and for integration differentiate one function and integrate another one till the reducing function becomes zero.

Answer 3

let me attach its solution too.. wait for a minute.

Answer 4

Like this: , it's x^4 so we can find I think the degree of x ? not sure what is called in English , but it's like this :

(ax^4 + bx^3 + cx^2 + dx + e)e^x

so we find the differentation:

(4ax^3+3bx^2+2cx+d)e^x + (ax^4 + bx^3 + cx^2 + dx + e)e^x

now:

x^4 = a
x^3=4a+b
x^2=3b+c
x=2c+d
+d+e

so:

a+1 will give us a=-1 , and b=4 , c=-12 , d=24 , e=-24

?? is this the right way ?

Answer 5

wow , that's much easier , thank you ^^

Answer 6

your welcome :)