Trigonometric integrals and substitutions E. Sköldberg [email protected] http://www.maths.nuigalway.ie/~emil/ School of Mathematics etc. National University of Ireland, Galway MA100 Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 1/1 Trigonometry Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 2/1 Trigonometric Identities Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 3/1 Odd power of cos Evaluate Z cos3 x dx Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 4/1 Odd power of sin Find Z sin3 x cos4 x dx Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 5/1 Even power of sin and cos Find Z sin2 x dx Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 6/1 Even power of sin and cos Find Z sin2 x cos2 x dx Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 7/1 Strategy To evaluate Z sinm x cosn x dx If m is odd, save one power of sin and convert the rest to cos using sin2 x = 1 − cos2 x If n is odd, save one power of cos and convert the rest to sin using cos2 x = 1 − sin2 x If both n and m are even, use the half-angle identities 1 sin2 x = (1 − cos 2x) 2 1 cos2 x = (1 + cos 2x) 2 Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 8/1 More useful identities We have for all angles A and B: sin A cos B = 12 (sin(A − B) + sin(A + B) sin A sin B = 12 (cos(A − B) − cos(A + B) cos A cos B = 12 (cos(A − B) + cos(A + B) Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 9/1 To evaluate integrals of the form Z sin mx cos nx dx we can use the identities from the previous slide. Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 10 / 1 Example Evaluate Z sin 4x cos 5x dx Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 11 / 1 Example Evaluate Z sin 4x cos 5x dx Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 11 / 1 Example Evaluate Z sin 4x cos 5x dx Sköldberg (National University of Ireland, Galway) Trigonometric integrals and substitutions MA100 11 / 1
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