Barrons AP Calculus - David Bock

The method yields

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.

E. APPLICATIONS OF ANTIDERIVATIVES; DIFFERENTIAL

EQUATIONS

The following examples show how we use given conditions to determine constants of integration.

EXAMPLE 48

Find f (x) if f ′(x) = 3x^2 and f (1) = 6.

SOLUTION:

Since f (1) = 6, 1^3 + C must equal 6; so C must equal 6 − 1 or 5, and f (x) = x^3 + 5.

EXAMPLE 49

Find a curve whose slope at each point (x, y) equals the reciprocal of the x-value if the curve

contains the point (e, −3).

SOLUTION: We are given that and that y = −3 when x = e. This equation is also solved

by integration. Since

Thus, y = ln x + C. We now use the given condition, by substituting the point (e, −3), to determine