Use integration by parts twice and treat the integral as the unknown. Then you can solve for the integral as if it's an algebra problem.
What we are trying to find is
If we find the derivative of I, we get
In reverse, we get that
and with a bit of rearranging we find
With me so far?
Now, if we were to divide both sides of the equation by 7, we'd have another expression for I (the function we are integrating). But there's another integral... Let's use the same process to solve for this integral.
Let's call this integral J...
so if .
Taking the derivative of J, we get
or in reverse, that
With a bit of rearranging we find that...
Substituting J into our original equation gives...
which, when expanded and manipulated, reduces to
And finally, we can solve for I, which is what we were finding...
(the integration constant).
Yes, it is a fair bit of writing, but is extremely powerful if all else fails...