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 - cos 2 x ) , cos 2 x = 1 2 (1 + cos 2 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 x dx , including cases in which m = 0 or n = 0, i.e.: ± cos n x dx ; ± sin m x dx . 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 x dx = ··· ( u = sin x , du = cos x dx ) ··· = ± 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 x dx = ± sin 2 x cos 2 x cos x dx = ± sin 2 x (1 - sin 2 x ) cos x dx = ··· ( u = sin x , du = cos x dx ) ··· = ± u 2 (1 - u 2 ) du = ± ( u 2 - u 4 ) du = u 3 3 - u 5 5 + C = sin 3 x 3 - sin 5 x 5 + C . If all the exponents are even then we use the half-angle identities.
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tri-nw - 1.6 TRIGONOMETRIC INTEGRALS AND TRIG SUBSTITUTIONS...

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