Math 20B Integral Calculus Lecture 12 ' 1 $ Tricks for trigs Slide 1 • Review: Recursion formulas. • Integral of some trigonometric functions. • integrals of f : IR → C. & % ' $ Reduction formulas are a simple way to write complicated integrals In the case of the function sin(x) one has: Slide 2 Z 1 (n − 1) (sin(x))n dx = − (sin(x))(n−1) cos(x) + n n Z (cos(x))n dx = & 1 (n − 1) (cos(x))(n−1) sin(x) + n n Z Z (sin(x))(n−2) dx. (cos(x))(n−2) dx. % Math 20B Integral Calculus Lecture 12 ' 2 $ Tricks for sines and cosines Slide 3 Consider integrals of the form Z [sin(x)]m [cos(x)]n dx. If m or n is odd, then the integral can be done by substitution, recalling: sin0 (x) = cos(x), cos0 (x) = − sin(x), sin2 (x) + cos2 (x) = 1. & % ' $ Tricks for sines and cosines Consider integrals of the form Z [sin(x)]m [cos(x)]n dx. Slide 4 If both m and n are even, then integral above can be done recalling 1 sin2 (x) = [1 − cos(2x)], 2 1 cos2 (x) = [1 + cos(2x)]. 2 & % Math 20B Integral Calculus Lecture 12 ' 3 $ Tricks for tangents and secants Slide 5 Consider integrals of the form Z [tan(x)]n [sec(x)]m dx. If n is odd or if m is even, then substitution recalling tan0 (x) = sec2 (x), sec0 (x) = tan(x) sec(x), sec2 (x) − tan2 (x) = 1. & % ' $ More tricks on sines and cosines The following integrals can be done with the associated formula: Slide 6 Z Z Z sin(mx) cos(nx) dx → sin(mx) sin(nx) dx → cos(mx) cos(nx) dx → & 1 [sin(α − β) + sin(α + β)]; 2 1 sin(α) sin(β) = [cos(α − β) − cos(α + β)]; 2 1 cos(α) cos(β) = [cos(α − β) + cos(α + β)]. 2 sin(α) cos(β) = % Math 20B Integral Calculus Lecture 12 ' 4 $ Integration can be generalized to functions f : IR → C These functions have the form f (x) = f1 (x) + i f2 (x), with f1 (x) and f2 (x) real functions. Slide 7 Examples: f (x) = 2x + i ln(x), 1 f 0 (x) = 2 + i , x g(x) = tan(3x) + i e2x g 0 (x) = 3 sec2 (3x) + 2i e2x . & % ' $ Riemann sums, integration, and the FTC can be generalized to function f : IR → C Z Slide 8 Example: f (x)dx = Z Z f1 (x)dx + i Z f2 (x)dx. 1 iax i e = − eiax . ia a Z 1 [x2 + i cos(x)]dx = x3 + i sin(x). 3 & eiax dx = % Math 20B Integral Calculus Lecture 12 ' Complex tricks for sine and cosine 5 $ Euler formulas eix = cos(x) + i sin(x), e−ix = cos(x) − i sin(x), Slide 9 imply that 1 ix [e + e−ix ], 2 1 ix sin(x) = [e − e−ix ]. 2i cos(x) = Therefore, integrals in sines and cosines can be transformed into complex integrals for exponentials. & %