Trigonometric Integrals - Bruce E. Shapiro

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Math 150A TOPIC 7. TRIGONOMETRIC INTEGRALS Hence ∫ sec 3 x dx = 1 (sec x tan x + ln | sec x + tan x|) + C 2 (7.24j) Now we collect some rules for products of secants and tangents. ∫ tan m x sec n x dx 1. If the exponent on the tangent is even (a) Save one factor of sec 2 x to use as d tan x. (b) Substitute sec 2 x = 1 + tan 2 x in the remaining secants. (c) Use a substitution u = tan x and integrate. 2. Otherwise, if the exponent of the secant is odd and the exponent of the tangent is at least 1, (a) Save one factor sec x tan x to use as d sec x. (b) Substitute tan 2 x = sec 2 x − 1 for all the remaining tangents. (c) Use a substitution u = sec x and integrate. 3. Otherwise, look for some other way to simplify the integral. Finally, we also consider simple products of multiple-angle formulas, which we demonstrate by example. Example 7.11 ∫ π/4 0 cos(2x) sin(12x) dx We use the trigonometric identity sin θ cos ψ = 1 [sin(θ − ψ) + sin(θ + ψ)] 2 (7.25a) to obtain ∫ π/4 0 ∫ π/4 cos(2x) sin(12x) dx = 1 [sin(10x) + sin(14x)] dx (7.25b) 2 0 = 1 [ ]∣ −1 −1 ∣∣∣ π/4 cos(10x) + 2 10 14 cos(14x) (7.25c) 0 = 1 [ −1 2 10 cos 5π 2 − 1 14 cos 7π 2 + 1 10 + 1 ] 14 (7.25d) = 3 35 (7.25e) Page 28 « 2012. Last revised: February 26, 2013
TOPIC 7. TRIGONOMETRIC INTEGRALS Math 150A Exercises ∫ 7.1. cos 2 x dx ∫ 7.2. cos 4 (3x) dx ∫ 7.3. sin 3 (5x) dx ∫ 7.4. cos 5 (x + 7) dx ∫ π/2 7.5. sin 2 (3x) dx 0 ∫ π 7.6. cos 4 x dx 0 ∫ 7.7. sin 2 x cos 2 x dx ∫ 7.8. sin x cos 3 x dx ∫ 7.9. sin 4 x cos 5 x dx ∫ 7.10. tan 2 (πx)dx 7.11. 7.12. 7.13. 7.14. 7.15. 7.16. 7.17. 7.18. 7.19. 7.20. ∫ 1 −1 tan 4 (πx/4)dx ∫ sec x tan 2 x dx ∫ sec 3 x tan x dx ∫ π/4 sec 3 x tan 3 x dx 0 ∫ sec x tan 3 x dx ∫ sec 7 x tan 9 x dx ∫ sec 4 x tan 7 x dx ∫ sec 4 x tan 4 x dx ∫ π/2 ∫ 0 sin(3x) sin(6x) dx sin(100x) cos(101x) dx « 2012. Last revised: February 26, 2013 Page 29
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