Summary of Trig Integration We now know the following facts about trig functions and calculus: sec x = 1 cos x tan x = sin x cos x sin2 x + cos2 x = 1 csc x = 1 sin x cot x = cos x sin x sec2 x = 1 + tan2 x d dx d dx tan x = sec2 x d dx d dx sin x = cos x � sec x = sec x tan x cos x = − sin x � tan x dx = − ln | cos(x)| + c sec x dx = ln(sec x + tan x) + c We’ve also seen several useful integration techniques, including methods for integrating any function of the form sinn x cosm x. At this point we have the tools needed to integrate most trigonometric polynomials. � Example: sec4 x dx We can get rid of some factors of sec x using the identity sec2 x = 1 + tan2 x. This is a particularly good idea because sec2 x is the derivative of tan x. � sec4 x dx = � � = (1 + tan2 x) sec2 x dx (1 + tan2 x) sec2 x dx. Using the substitution u = tan x, du = sec2 x dx, we get: � � sec4 x dx = (1 + u2 )du = � sec4 x dx = u3 +c 3 tan3 x tan x + + c. 3 u+

Summary of Trig Integration

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