"All limits, especially national ones, are contrary to the nature of mathematics… Mathematics knows no races… For mathematics the whole cultural world is a single country" – David Hilbert. "Face problems with a minimum of blind calculation, a maximum of seeing thought" – Hermann Minkowski

Integration by parts – video clips

First take 4 minutes to watch:

That refers to the LIATE rule:

It’s a rule of thumb proposed by Herbert Kasube: whichever function comes first in the following list should be u:

L – Logarithmic functions: ln x, log_{b} x, etc.
I – Inverse trigonometric functions: arctan x, arcsec x, etc.
A – Algebraic functions: x^{2}, 3x^{50}, etc.
T – Trigonometric functions: sin x, tan x, etc.
E – Exponential functions: e^{x}, 19^{x}, etc.

The function which comes lower in the list should be dv.

Functions lower on the list have easier integrals than the functions above them, and functions higher on the list are more likely to produce derivatives simpler than themselves.

It’s not infallible, but it should do for C4.

And then, for a worked example:

And then, for more on calculus from students at Townsend Harris High School in New York: