Integration problem

**Mourits** · June 9th 2010, 07:36 AM

Can anyone help me to get $S(x)$ from

$<br /> S^{'} (x)=x^{-b} e^{ -a x}<br />$
$x>0$

If I tryed to use integration by parts, by I can't figure put, what to call $f$ and $g$ . And Maple does not give me anything to go with.

**redsoxfan325** · June 9th 2010, 08:41 AM

Originally Posted by Mourits

Can anyone help me to get $S(x)$ from

$<br /> S^{'} (x)=x^{-b} e^{ -a x}<br />$
$x>0$

If I tryed to use integration by parts, by I can't figure put, what to call $f$ and $g$ . And Maple does not give me anything to go with.

If it's a definite integral (from $0$ to $\infty$ ), then the answer is $a^{b-1}\Gamma(1-b)$ , and I can show that, but I'm not sure how to solve the indefinite integral to find $S(x)$ . Wolfram Alpha gives this answer, but I don't know how to get there at the moment.

**Deadstar** · June 9th 2010, 09:57 AM

Originally Posted by redsoxfan325

If it's a definite integral (from $0$ to $\infty$ ), then the answer is $a^{b-1}\Gamma(1-b)$ , and I can show that, but I'm not sure how to solve the indefinite integral to find $S(x)$ . Wolfram Alpha gives this answer, but I don't know how to get there at the moment.

I think your definite integral answer is the one he is after but... As for the indefinite one.

Express $e^{-ax}$ as a power series and integrate etc...

Should end up with...

$x^{b+1} \sum_{n=0}^{\infty} \frac{(-ax)^n}{(b+n+1)n!}$

But I'm not sure how to complete it from there!

**drumist** · June 9th 2010, 10:08 AM

Supposing we are actually interested in the definite integral... in case Mourits needs to see how to arrive at that answer.

By letting $x=t$ and $a=s$ , we can see this is just a Laplace transform:

$\int_0^{\infty} x^{-b} e^{-ax} \, dx = \int_0^{\infty} t^{-b} e^{-st} \, dt = \mathcal{L} \{ t^{-b} \}$

Using a Laplace transform table, we get

$\mathcal{L} \{ t^{-b} \} = \frac{\Gamma(-b+1)}{s^{-b+1}} = \frac{\Gamma(1-b)}{a^{1-b}} \qquad \mbox{for }b<1 \mbox{ and } a>0$

**Mourits** · June 9th 2010, 10:46 AM

Thanks

Thread: Integration problem

LinkBack

Thread Tools

Display

Integration problem

Similar Math Help Forum Discussions

integration problem

Simple Integration problem: Trigonometric Integration

Integration Problem

Integration problem !!

Integration problem

Search Tags