Integration by Parts

Today in Calculus we learned a new method for integration known as integration by parts. This method is useful for solving antiderivatives of functions that are combinations (two functions multiplied together), which cannot be solved by regular u substitution.

if uv = something then
d(uv) = udv + vdu (product rule)
uv = fint (udv) + fint(vdu)
uv - fint(vdu) = fint(udv)

So the idea is to take your function and find the most complicated part of the function (usually this is the highest order function in the function hierarchy, but it can also be the most complex sometimes like a trig function). This complicated part must have a known antiderivative (that you can do by integration rules). The "left over" part is the v. Solve for dv and u. You then set up a table:

v
dv
u
du

Finally, when you have identified the elements/ finished the table you plug in the appropriate expressions in the correct place in the equation above. Sometimes when evaluating a 'simplified' expression, you may come across an antiderivative of another combination function (aka the fint(vdu) or fint(udv) cannot be automatically solved). This could mean that you may have set up the table wrong but it can also mean that further integration by parts is necessary and that your work is at an intermediate step to the ultimate simplified form.