- #1
Shinjo
- 12
- 0
I'm having a tough time trying to do integration by parts with one of my limits being infinity. My Integral looks like:
[tex] \int_0^\infty x^z e^{-x} dx[/tex] with [tex] z = \frac{-1}{\pi}[/tex]
Now if I let [tex]u = e^{-x} [/tex] and [tex]dv = x^z dx[/tex],
I will have: [tex]du = -e^{-x} dx [/tex] and [tex] v = \frac{1}{z + 1} x^{z + 1}
[/tex] and so
[tex] uv - \int_0^\infty v du
= e^{-x} \frac{1}{z + 1} x^{z + 1} - \int_0^\infty \frac{1}{z + 1} x^{z + 1} -e^{-x} dx[/tex]
However, since one of the limit is infinity, the term
[tex] uv = e^{-x} \frac{1}{z + 1} x^{z + 1}[/tex] has a freakin infinity subbed in it. The answer is actually,
[tex] \frac{1}{z + 1} \int_0^\infty x^{z + 1} e^{-x} dx[/tex], which is what my integral part looks like. What am I doing wrong? Why do I get a term with infinity at the front?
[tex] \int_0^\infty x^z e^{-x} dx[/tex] with [tex] z = \frac{-1}{\pi}[/tex]
Now if I let [tex]u = e^{-x} [/tex] and [tex]dv = x^z dx[/tex],
I will have: [tex]du = -e^{-x} dx [/tex] and [tex] v = \frac{1}{z + 1} x^{z + 1}
[/tex] and so
[tex] uv - \int_0^\infty v du
= e^{-x} \frac{1}{z + 1} x^{z + 1} - \int_0^\infty \frac{1}{z + 1} x^{z + 1} -e^{-x} dx[/tex]
However, since one of the limit is infinity, the term
[tex] uv = e^{-x} \frac{1}{z + 1} x^{z + 1}[/tex] has a freakin infinity subbed in it. The answer is actually,
[tex] \frac{1}{z + 1} \int_0^\infty x^{z + 1} e^{-x} dx[/tex], which is what my integral part looks like. What am I doing wrong? Why do I get a term with infinity at the front?