Calculus II: Dr. Staples
Section 7.2 Trigonometric Integrals
Pythagorean Formulas
𝑠𝑖𝑛! 𝑥 + 𝑐𝑜𝑠 ! 𝑥 = 1
Half-Angle Formulas
1 − cos 2𝑥
𝑠𝑖𝑛! 𝑥 = 2
1
+
cos
2𝑥
𝑐𝑜𝑠 ! 𝑥 = 2
𝑡𝑎𝑛! 𝑥 + 1 = 𝑠𝑒𝑐 ! 𝑥
𝑐𝑜𝑡 ! 𝑥 + 1 = 𝑐𝑠𝑐 ! 𝑥
Powers of Sine and Cosine Strategy:
𝐬𝐢𝐧𝒎 𝒙 𝐜𝐨𝐬 𝒏 𝒙 𝒅𝒙
1. m is odd, n is a real number
2. n is odd, m is a real number
3. m, n ≥ 0, both even
1. Split off one power of sin x, rewrite the remaining even
number of powers of sin x in terms of cos x. Use a u-sub
with u = cos x.
2. Split off one power of cos x, rewrite the remaining even
number of powers of cos x in terms of sin x. Use a u-sub
with u = sin x.
3. Use half-angle formulas to transform the integrand into a
polynomial in cos 2x. Repeat former strategies as needed.
Integrals of tan x, cot x, sec x, and csc x
𝑡𝑎𝑛 𝑥 𝑑𝑥 = −𝑙𝑛 𝑐𝑜𝑠 𝑥 + 𝐶 = 𝑙𝑛 𝑠𝑒𝑐 𝑥 + 𝐶
𝑐𝑜𝑡 𝑥 𝑑𝑥 = 𝑙𝑛 𝑠𝑖𝑛 𝑥 + 𝐶
𝑠𝑒𝑐 𝑥 𝑑𝑥 = 𝑙𝑛 𝑠𝑒𝑐 𝑥 + 𝑡𝑎𝑛 𝑥 + 𝐶
𝑐𝑠𝑐 𝑥 𝑑𝑥 = −𝑙𝑛 𝑐𝑠𝑐 𝑥 + 𝑐𝑜𝑡 𝑥 + 𝐶
Powers of tan x and sec x Strategy:
𝐭𝐚𝐧𝒎 𝒙 𝐬𝐞𝐜 𝒏 𝒙 𝒅𝒙
1. n is even
1. Split off sec2x, rewrite the remaining even number of
powers of sec x in terms of tan x. Use a u-sub with u = tan x.
2. n is odd, m is odd
2. Split off (sec x tan x), rewrite the remaining even number
of powers of tan x in terms of sec x. Use a u-sub with u=sec x.
3. m even and n odd
Change all to powers of sec x. Be able to integrate sec 3 x. Do
not worry about higher powers of sec x.