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Trigonometric Integrals - Section 7.2 Integrals Involving Powers of Sine and Cosine: ∫ sin m x cos n x dx Useful trigonometric identities: sin 2 x + cos 2 x = 1 tan 2 x + 1 = sec 2 x sin 2x = 2 sin x cos x sin 2 x = 1 1 − cos 2x 2 1 + cot 2 x = csc 2 x cos 2x = cos 2 x − sin 2 x cos 2 x = 1 1 + cos 2x 2 (Pythagorean Identities) (Double Angle Identities) (Power Reducing Formulas) Use a trigonometric identity to evaluate the following integral: ∫ sin 2 x dx Now evaluate the integral ∫ sin 3 x dx 1 Reduction Formulas for Integrals Involving Powers of Sine and Cosine We can prove these using integration by parts ∫ sin n x dx = − 1n sin n−1 x cos x + ∫ cos n x dx = n−1 n 1 cos n−1 x sin x + n − 1 n n ∫ sin n−2 x dx ∫ cos n−2 x dx General Guidelines for evaluating integrals of the form ∫ sin m x cos n x dx 1. If the power of sine is odd and positive, save one sine factor and convert the remaining factors to cosine by the Pythagorean Identity. Then, expand and integrate by substitution choosing u = cos x. ∫ sin Odd 2k+1 Convert to cosine ∫ x cos n x dx = sin 2 x k Save for du cos n x sin x dx = ∫1 − cos 2 x k cos n x sin x dx 2. If the power of cosine is odd and positive, save one cosine factor and convert the remaining factors to sine by the Pythagorean Identity. Then, expand and integrate by substitution choosing u = sin x. ∫ sin m x cos Odd 2k+1 Convert to sine x dx = ∫ sin m x cos 2 x k Save for du cos x dx = ∫ sin m x1 − sin 2 x k cos x dx 3. If the powers of both sine and cosine are even and nonnegative, make repeated use of the identities sin 2 x = 1 − cos 2x and cos 2 x = 1 + cos 2x 2 2 to convert the integrand to odd powers of the cosine. Then proceed as in guideline #2. Alternatively, if m ≤ n, use the identity sin 2 x = 1 − cos 2 x to replace sin m x by 1 − cos 2 x m/2 to obtain ∫ sin m x cos n x dx = ∫1 − cos 2 x m/2 cos n x dx and expand the integral on the right to obtain a sum of integrals of powers of cos x and use the reduction formula ∫ cos n x dx = If m > n, replace cos n x by 1 − sin 2 x n/2 1 cos n−1 x sin x + n − 1 n n ∫ cos n−2 x dx to obtain ∫ sin m x cos n x dx = ∫ sin m x1 − sin 2 x n/2 dx and expand the integral on the right to obtain a sum of integrals of powers sin x and evaluate using the reduction formula ∫ sin n x dx = − 1n sin n−1 x cos x + n−1 n ∫ sin n−2 x dx 2 Evaluate the following Integrals: 1. ∫ sin 2 x cos 3 x dx 2. ∫ sin 5 x cos 4 x dx 3 3. ∫ sin 6 x dx 4. ∫ sec x dx 4 Integrals Involving Powers of Tangent and Secant Recall these integral forms: ∫ tan x dx = − ln|cos x| + C = ln|sec x| + C ∫ sec x dx = ln|sec x + tan x| + C Use the following guidelines for integrals of the form ∫ sec m x tan n x dx (taken from Calculus by Larson, Hostetler, & Edwards, 6th edition): 1. If the power of the secant is even and positive, save a secant-squared factor (for the du) and convert the remaining factors to tangents. Then expand and integrate by substitution with u = tan x. Convert to tangents Even ∫ sec 2k x tan n x dx = ∫ sec 2 x Save for du tan n x sec 2 x dx= ∫1 + tan 2 x k−1 tan n x sec 2 x dx k−1 2. If the power of the tangent is odd and positive, save a secant-tangent factor (for the du) and convert the remaining factors to secants. Then expand and integrate by substitution with u = sec x. Convert to secants Odd ∫ sec m x tan 2k+1 x dx = ∫ sec m−1 x tan 2 x k Save for du sec x tan x dx= ∫ sec m−1 xsec 2 x − 1 k sec x tan x dx Evaluate the following: For #’s 1 & 2 use the substitution u = tan x and reserve a sec 2 x factor for the "du" 1. ∫ tan 2 x sec 2 x dx 5 2. ∫ tan 2 x sec 4 x dx Let u = sec x and reserve a sec x tan x factor for the "du" 1. ∫ tan 3 x sec 3 x dx 6