More Integrals involving Trigonometric Functions

Trig

Integration with Trig Functions Example 3: I1 = ∫ tan^m x sec^n x dx with m an odd integer and n an even integer; i.e. I1 = ∫ tan^3 x sec^2 x dx Now let u = tan x du = sec^2x dx So I1 = ∫ u^3 du Similar reasoning for the integral: I1a = ∫ cot^m x csc^n x dx Example 4: (much harder) I2 = ∫ tan^m x sec^n x dx with m an even integer and n an odd integer; i.e. I2 = ∫ tan^2 x sec^3 x dx tan^2x = sec^2 x - 1 I2 = ∫ (1 + sec^2 x) sec^3 x dx I2 = ∫ (sec^5 x - sec^3 x)dx Next use integration by parts on each of these 2 integrals; first, I2a = ∫sec^3 x dx Next let u = sec x dv = sec^2x dx du = sec x tan x dx v = tan x I2a = ∫sec^3 x dx = sec x tan x - ∫sec x tan^2 x dx Click here to continue with Example 4.

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