tri-nw - 1.6 TRIGONOMETRIC INTEGRALS AND TRIG SUBSTITUTIONS...

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1.6. TRIGONOMETRIC INTEGRALS AND TRIG. SUBSTITUTIONS 26 1.6. Trigonometric Integrals and Trigonometric Substitutions 1.6.1. Trigonometric Integrals. Here we discuss integrals of pow- ers of trigonometric functions. To that end the following half-angle identities will be useful: sin 2 x = 1 2 (1 - cos2 x ) , cos 2 x = 1 2 (1 + cos2 x ) . Remember also the identities: sin 2 x + cos 2 x = 1 , sec 2 x = 1 + tan 2 x. 1.6.1.1. Integrals of Products of Sines and Cosines. We will study now integrals of the form sin m x cos n xdx, including cases in which m = 0 or n = 0, i.e.: cos n xdx ; sin m xdx. The simplest case is when either n = 1 or m = 1, in which case the substitution u = sin x or u = cos x respectively will work. Example : sin 4 x cos xdx = · · · ( u = sin x , du = cos xdx ) · · · = u 4 du = u 5 5 + C = sin 5 x 5 + C . More generally if at least one exponent is odd then we can use the identity sin 2 x +cos 2 x = 1 to transform the integrand into an expression containing only one sine or one cosine.
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1.6. TRIGONOMETRIC INTEGRALS AND TRIG. SUBSTITUTIONS 27 Example : sin 2 x cos 3 xdx = sin 2 x cos 2 x cos xdx = sin 2 x (1 - sin 2 x )cos xdx = · · · ( u = sin x , du
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