− cos x. Then,EXAMPLE 46
Find
SOLUTION: We let u = ln x and dv = x^4 dx. Then, and Thus,THE TIC-TAC-TOE METHOD.^1
This method of integrating is extremely useful when repeated integration by parts is necessary. To
integrate we construct a table as follows:
Here the column at the left contains the successive derivatives of u(x). The column at the right
contains the successive antiderivatives of v(x) (always with C = 0); that is, v 1 (x) is the antiderivative
of v(x), v 2 (x) is the antiderivative of v 1 (x), and so on. The diagonal arrows join the pairs of factors
whose products form the successive terms of the desired integral; above each arrow is the sign of that
term. By the tic-tac-toe method,
EXAMPLE 47
To integrate cos x dx by the tic-tac-toe method, we let u(x) = x^4 and v(x) = cos x, and get the
following table: