integration by parts/substitution

[cheffy]

integral of e^(-sqrt(x))

I tried substituting and integrating by parts and I'm stuck. Any suggestions?

Thanks!

 

[galactus]

Let \(\displaystyle \L\\u=\sqrt{x}, \;\ u^{2}=x, \;\ 2udu=dx\)

\(\displaystyle \L\\2\int{ue^{-u}}du\)

 

[mark07]

Use the substitution

\(\displaystyle \L y^2 = x \text{ with } y>0\)

\(\displaystyle \L 2y dy = dx\)

Then your integral becomes

\(\displaystyle \L \int e^{-y} 2 y dy = 2 \int y e^{-y} dy\)

Now use by parts with \(\displaystyle u=y\) and \(\displaystyle dv = e^{-y} dy\) ...

Oops.. Galactus was faster than me, never mind...

 

[skeeter]

let \(\displaystyle \L t = -\sqrt{x}\)

\(\displaystyle \L dt = -\frac{1}{2\sqrt{x}} dx\)

\(\displaystyle \L dx = -2\sqrt{x} dt = 2t dt\)

\(\displaystyle \L \int e^{-\sqrt{x}} dx\)

substitute ...

\(\displaystyle \L \int 2t e^t dt\)

integration by parts yields the antiderivative ...

\(\displaystyle \L 2t e^t - 2e^t + C = 2e^t(t - 1) + C\)

back-substitute ...

\(\displaystyle \L -2e^{-\sqrt{x}}(\sqrt{x} + 1) + C\)