Download Section 8.3 - Trigonometric Integrals

Section 8.3 - Trigonometric Integrals
GROUP I:
Z
sin x dx,
n
Z
cosn x dx
METHOD: Use the following identities:
Z
Z
n−1
1
n−1
n
x cos x +
sinn−2 x dx, n ≥ 2
sin x dx = − sin
n
n
Z
Z
n−1
1
n−1
n
x sin x +
cos x dx = cos
cosn−2 x dx, n ≥ 2
n
n
Z
GROUP II:
sinm x cosn x dx
METHOD:
(a) If n is odd, then u = sin x and use cos2 x = 1 − sin2 x
(b) If m is odd, then u = cos x and use sin2 x = 1 − cos2 x
(c) If n and m are even, then use the identities
1
1
cos2 x = (1 + cos 2x)
sin2 x = (1 − cos 2x),
2
2
Z
Z
Z
GROUP III:
sin mx cos nx dx,
sin mx sin nx dx,
cos mx cos nx dx
METHOD: Use the following identities:
1
sin α cos β = [sin(α − β) + sin(α + β)]
2
1
sin α sin β = [cos(α − β) − cos(α + β)]
2
1
cos α cos β = [cos(α − β) + cos(α + β)]
2
GROUP IV:
Z
tan x dx,
n
Z
secn x dx
METHOD: Use the following identities:
Z
tan x dx = ln | sec x| + C
Z
Z
Z
sec x dx = ln | sec x + tan x| + C
tann−1 x
−
tan x dx =
n−1
n
Z
tann−2 x dx,
secn−2 x tan x n − 2
+
sec x dx =
n−1
n−1
n
Z
n≥2
secn−2 x dx,
n≥2
GROUP V:
Z
tanm x secn x dx
METHOD:
(a) If n is even, then u = tan x and use sec2 x = tan2 x + 1
(b) If m is odd, then u = sec x and use tan2 x = sec2 x − 1
(c) If n is odd and m is even, then use tan2 x = sec2 x − 1