Integration Revision – Integration By Substitution

Now, looking at the other integral from Oct 13th...

$\int^{x}_{0}4xe^{-2x^{2}}\:dx=1-e^{-2x^{2}}$

In my opinion, this was scarier than the first, but looking at it again, what makes it scary is the various powers. e to the power of x to the power of 2 and such.

But why should this be scary? There's practically only one tool that one could use to solve this: the only tool that includes functions of functions, integration by substitution:

$\int f(g(x))\:g^{\prime}(x)\:dx=\int f(u)\: du$

In our case, the inner function:
$g(x)=-2x^{2}=u$
so:
$g^{\prime}(x)=-4x$

and the outer function:
$f(x)=e^{x}$

Plugging all these into our "integration by substitution" tool gives:

$\int -4xe^{-2x^{2}}\:dx$

But this is slightly different to the original integral due to that minus sign. Of course this is trivial to deal with as:

$\int^{x}_{0}4xe^{-2x^{2}}\:dx=-\int^{x}_{0} -4xe^{-2x^{2}}\:dx$

Great! So now we have the structure we require to apply integration by substitution! We can substitute all of that with $f(u)$ , and all the scary bits go away. So: