# How do you evaluate the definite integral #int_0^(oo)x^3e^(-x)dx#?

##### 2 Answers

first, integrate the indefinite integral

with some integration by parts, you should get

the answer is

since

so the answer is

See below.

#### Explanation:

Filling in some of the extra steps...

You can use integration by parts.

#uv-intvdu#

Let:

#u=x^3 " " du=3x^2dx#

#dv=e^(-x)" " v=-e^(-x)#

Substituting into the above expression:

#=>-x^3e^(-x)+3intx^2e^(-x)dx#

Apply again:

#u=x^2 " " du=2xdx#

#dv=e^(-x)" " v=-e^(-x)#

#=>-x^3e^(-x)+3[-x^2e^(-x)+2intxe^(-x)dx]#

Finally:

#u=x " " du=dx#

#dv=e^(-x)" " v=-e^(-x)#

#=>-x^3e^(-x)+3[-x^2e^(-x)+2{-xe^(-x)+inte^(-x)dx}]#

Simplify:

#-e^(-x)(x^3+3x^2+6x+6)#

Evaluate from

Note that *very* big number (to say the least). This is approximately **zero**.

#=>0-[-1/e^0((0)^3+3(0^2)+6(0)+6)]#

Note that

#=>1(6)=6#