Integrals Involving Inverse Trigonometric Functions. The derivatives of the Inverse trig functions fall into three pairs, where one is just the negative of its paired function. Arcsin and arccos pair up, arctan and arccot pair up, and arcsec and arccsc pair up. The main three rules are below.
Ex 1 p. 380 Integration with Inverse Trig Functions To make the form fit, a = 3 and u = 2x. Then du = 2dx This looks like an arcsin form
Ex 1 continued a = 3, u = 2x, du = 2dx
Ex 2 p.381 Integration by Substitution Now it fits the arc sec rule
Ex 3 p.381 Rewriting the sum of two quotients Doesn’t really fit one of the rules as is. Rewrite to use power rule and arcsin rule
Completing the Square: Helps when quadratic function is involved in the integrand. Just to remind you of the process, start with lead coefficient of 1. Ex 4 p. 382 Completing the square Looking through our possible rules, we see arctan has sum of two squares in denominator.
If the lead coefficient is not one, factor it out before completing the square. This also works if it is a negative lead coefficient.
Ex 6 p. 382 Completing the square (negative lead coefficient) Solution:
Notice #4,5 not in front of book
Ex 6 p. 382 Comparing integration problems. Looking at derivative rules, you now have a way to differentiate any elementary function. With integration this is far from true. We have to make things fit the right form or we can’t integrate. Fits arcsec rule Fits power rule Doesn’t fit any rule so far – see p. A22, #31
Ex 7 p. 384 Comparing Integration problems Let u = lnx then du=(1/x)dx Let u = lnx then du = (1/x)dx We don’t have a rule for this – p.A27 #87 Notice that in both Ex 6 and Ex 7, the simplest ones are the hardest to find rules for.
5.7 p. 385/ 1-45 every other odd, 53-54, 63-69 odd, 75-77, 83