2009
Montgomery
Inverse Spring Substitution
What to do about integrals that have a2 x2 on the inside? We have seen with integrals containing a square root and some combination of a2 and x2 , a inverse substitution will make the integral possible to solve. This is a substitution of the form x = a trig. . We use this substitution in order to use properties of trigonometry, summarized in the following chart: Expression a2 x2 a2 + x2 x2 a2 Substitution x = a sin x = a tan x = a sec Identity 1 sin2 = cos2 1 + tan2 = sec2 sec2 1 = tan2
Keep in mind, unlike regular substitution, this actually makes the integral MORE COMPLICATED! However, by using trig. identities, we can simplify these integrals considerably; to usually an integral involving trigonometry. This can take several steps, and several techniques of integration. Eventually, we get an answer like: a2 x2 dx = H() + C x How do we get an answer in terms of x? By using our original substitution x = a trig. . trig. = and a Right Triangle Trigonometry.
sin =
opp. a = hyp. c hyp. c = opp. a
cos =
adj. b = hyp. c hyp. c = adj. b
tan =
opp. a = adj. b adj. b = opp. a
cs...