Calculus AB and Calculus BC

CHAPTER 5 Antidifferentiation

D. INTEGRATION BY PARTS

Parts Formula

The Parts Formula stems from the equation for the derivative of a product:

Image or, or more conveniently d(uv) = u dv + v du.

Hence, u dv = d(uv) − v du and integrating gives us Image or

Image

the Parts Formula. Success in using this important technique depends on being able to separate a given integral into parts u and dv so that (a) dv can be integrated, and (b) Image du is no more difficult to calculate than the original integral.

EXAMPLE 43

Find Image

SOLUTION: We let u = x and dv = cos x dx. Then du = dx and v = sin x. Thus, the Parts Formula yields

Image

EXAMPLE 44

Find Image

SOLUTION: We let u = x2 and dv = ex dx. Then du = 2x dx and v = ex, so Image We use the Parts Formula again, this time letting u = x and dv = ex dx so that du = dx and v = ex. Thus,

Image

EXAMPLE 45

Find I = Image

SOLUTION: To integrate, we can let u = ex and dv = cos x dx; then du = ex dx, v = sin x. Thus,

Image

To evaluate the integral on the right, again we let u = ex, dv = sin x dx, so that du = ex dx and v = − cos x. Then,

Image

EXAMPLE 46

Find Image

SOLUTION: We let u = ln x and dv = x4 dx. Then, Image and Image Thus,

Image

THE TIC-TAC-TOE METHOD. 1

This method of integrating is extremely useful when repeated integration by parts is necessary. To integrate Image we construct a table as follows:

Image

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, v1 (x) is the antiderivative of v(x), v2 (x) is the antiderivative of v1 (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,

Image

EXAMPLE 47

To integrate Image cos x dx by the tic-tac-toe method, we let u(x) = x4 and v(x) = cos x, and get the following table:

Image

The method yields

Image

With the ordinary method we would have had to apply the Parts Formula four times to perform this integration.

1 This method was described by K. W. Folley in Vol. 54 (1947) of the American Mathematical Monthly and was referred to in the movie Stand and Deliver.