*can’t*do into a simple product minus an integral you

*can*do. Here’s the formula:

Don’t try to understand this yet. Wait for the examples that follow.

If you remember that, you can easily remember that the integral on the right is just like the one on the left, except with the *u* and *v* reversed.

Here’s the method in a nutshell.

First, you’ve got to split up the integrand into a *u* and a *dv* so that it fits the formula. For this problem, choose ln(*x*) to be your *u. *Then, everything else is the *dv,* namely

Next, you differentiate *u* to get your *du*, and you integrate *dv* to get your *v*. Finally, you plug everything into the formula and you’re home free.

*u*, ln(

*x*), in the upper-left corner and your

*dv*,

in the lower-right corner, as in the following figure.

Now complete the box:

The completed box for

is shown in the next figure.

A good way to remember the integration-by-parts formula is to start at the upper-left square and draw an imaginary number 7 — across, then down to the left, as shown in the following figure. This is an oh-so-sevenly mnemonic device (get it?—“sevenly” like “heavenly”—ha, ha, ha, ha.)

Remembering how you draw the 7, look back to the figure with the completed box. The integration-by-parts formula tells you to do the top part of the 7, namely

minus the integral of the diagonal part of the 7,

(By the way, this method is much easier to do than to explain. Try the box technique with the 7 mnemonic. You’ll see how this scheme helps you learn the formula and organize these problems.)

Ready to finish? Plug everything into the formula:

This is because –2/3 times any number is still just any number.

Here’s a great mnemonic device for how to choose the *u* (again, once you’ve selected your *u*, everything else is automatically the *dv*.

*LIATE*to help you choose your

*u*(calculus nerds can check out Herb’s article in the

*American Mathematical Monthly*90, 1983 issue):

L |
Logarithmic | like log(x) |

I |
Inverse trigonometric | like arctan(x) |

A |
Algebraic | like 5x + 3 (x can be raised to any power) |

T |
Trigonometric | like cos(x) |

E |
Exponential | like 10 raised to the x power |

*x*, not just a number.)