Page 210 - Calculus Workbook For Dummies

194 Part IV: Integration and Infinite Series

Q. Integrate # x 2 lnx dx. 4. Follow the arrows in the following box
to help you remember how to use the
A. 1 x c lnx - 1 m + C integration by parts formula.
3
3 3
1. Pick your u function.
The integrand contains a logarithmic
function (first on the LIATE list), so
ln x is your u. Everything else in the
2
integrand — namely x dx — is
automatically your dv.
2. Use a box like the one in the following
figure to organize the four elements of
the problem. Your original integral equals the product
of the two cells along the top minus the
integral of the product of the cells on the
diagonal. (Think of drawing a “7” —
u v that’s your order.)
x # 2 lnx dx = lnx $ 1 x - # c 1 x $ 1 m dx
3
3
3 3 x
du dv 5. Simplify and integrate.
1 3 lnx - # 2
1
= x x dx
3 3
1 1
Put your u and your dv in the appropri- = 1 x 3 lnx - $ x + C
3
ate cells, as the following figure shows. 3 3 3
1 3 1
= x c lnx - m + C
3 3
ln(x) You’re done.

2
x dx

3. Differentiate u and integrate dv, as the
arrows in the figure show.

ln(x) 1 x 3
3
diff int

1 dx x dx
2
x