1. ## algebraic/trigonometric substitution

$\displaystyle \int\frac{e^-x}{(9e^-2x+1)^3/2}\, dx$

$\displaystyle \int\frac{dz}{(z^2-6z+18)^3/2}$

$\displaystyle \int\frac{ln^3w}{w\sqrt{ln^2w-4}}\,dw$

2. Originally Posted by cazimi
$\displaystyle \int\frac{e^{-x}}{(9e^{-2x}+1)^{3/2}}\, dx$
Substitute $\displaystyle u=e^{-x},$

$\displaystyle \int {\frac{{e^{ - x} }} {{\left( {9e^{ - 2x} + 1} \right)^{3/2} }}\,dx} = - \int {\frac{{du}} {{\left( {9u^2 + 1} \right)\sqrt {9u^2 + 1} }}} .$

Make another substitution defined by $\displaystyle u=\frac1\tau,$

$\displaystyle \int {\frac{{e^{ - x} }} {{\left( {9e^{ - 2x} + 1} \right)^{3/2} }}\,dx} = \int {\frac{\tau } {{\left( {9 + \tau ^2 } \right)\sqrt {9 + \tau ^2 } }}\,d\tau } .$

One more substitution according to $\displaystyle \varphi = \sqrt {9 + \tau ^2 } ,$

$\displaystyle \int {\frac{{e^{ - x} }} {{\left( {9e^{ - 2x} + 1} \right)^{3/2} }}\,dx} = \int {\varphi ^{ - 2} \,d\varphi } = - \frac{1} {\varphi } + k.$

Back substitute:

\displaystyle \begin{aligned} \varphi &= \sqrt {9 + \frac{1} {{u^2 }}}\\ &= \sqrt {9 + \frac{1} {{e^{ - 2x} }}}\\ &= \frac{{\sqrt {9e^{ - 2x} + 1} }} {{e^{ - x} }}. \end{aligned}

And we happily get $\displaystyle \int {\frac{{e^{ - x} }} {{\left( {9e^{ - 2x} + 1} \right)^{3/2} }}\,dx} = - \frac{{e^{ - x} }} {{\sqrt {9e^{ - 2x} + 1} }} + k.$

3. ## Second one

On the second integral,

First, I think you meant
$\displaystyle \int\frac{dz}{(z^2-6z+18)^{3/2}}$, not $\displaystyle \int\frac{dz}{(z^2-6z+18)^3/2}$ (you forgot to bracket the exponent).

First, we want to complete the square in the denominator: $\displaystyle z^2-6z+18=z^2-6z+9+9=(z-3)^2+9$

So using $\displaystyle u=z-3$, we get $\displaystyle \int\frac{dz}{(z^2-6z+18)^{3/2}} = \int\frac{du}{(u^2+3^2)^{3/2}}$

From here, you can use the trigonometric substitution $\displaystyle u=3 \tan \theta$, which gives $\displaystyle u^2+3^2 = 3^2 (\tan^2 \theta +1) = (3 \sec \theta)^2$ and $\displaystyle du = 3 \sec^2 \theta d\theta$
And thus
$\displaystyle \int\frac{dz}{(z^2-6z+18)^{3/2}} = \int\frac{3 \sec^2 \theta d\theta}{(3 \sec \theta)^3} = \int\frac{d\theta}{9 \sec \theta} = \frac{1}{9} \int \cos \theta d\theta$

You can take it from there.

--Kevin C.

4. Well, the third

Let $\displaystyle u=\ln(w)$ thus: $\displaystyle \frac{du}{dw}=\frac{1}{w}$

Then $\displaystyle \int\frac{ln^3w}{w\sqrt{ln^2w-4}}dw=\int\frac{u^3}{\sqrt{u^2-4}}du$

Now $\displaystyle z=\sqrt{u^2-4}$ so: $\displaystyle \frac{dz}{du}=\frac{u}{\sqrt{u^2-4}}$

Therefore: $\displaystyle \int\frac{u^3}{\sqrt{u^2-4}}du=\int(z^2+4)dz$