Trigonometric Integrals - Bruce E. Shapiro

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