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:

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

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

=-\int^{u}_{0} e^{u}\:du

=-[ e^{u}]^{u}_{0}

=-(e^{u}-1)

=1-e^{u}

Then substituting u (=g(x)) back in:

=1-e^{-2x^{2}}