I do not understand the concept of reduction formula's at all.
How would I use a reduction formula to solve the following
$\displaystyle \int{x^4e^x}$
Is there something general to solve any kind of integral of this form.
I do not understand the concept of reduction formula's at all.
How would I use a reduction formula to solve the following
$\displaystyle \int{x^4e^x}$
Is there something general to solve any kind of integral of this form.
The reduction formula for this one is
$\displaystyle
\int x^n e^x\,dx = x^n e^x - n\int x^{n-1} e^x\,dx
$
so putting $\displaystyle n = 4$ gives
$\displaystyle
\int x^4 e^x\,dx = x^4 e^x - 4\int x^{3} e^x\,dx
$
now use the formula for $\displaystyle n=3$
$\displaystyle
\int x^4 e^x\,dx = x^4 e^x - 4\left(x^3 e^x - 3\int x^{2} e^x\,dx\right)
$
and keep going until you reach an integral like $\displaystyle \int e^x\,dx $ which you can do.