Math 2414 Section 7 3 Trigonometric Integrals Simple Strategies Substitution and multiplying by 1 are simple strategies often employed when evaluating trigonometric integrals Evaluate the following integral tan x dx Evaluate the following integral 4p 3 1 cos 2q dq p 4 Common Integrals Involving Substitution The following integrals require substituting or multiplying by 1 and then substituting In any case we will see these integrals frequently and should be memorized tan x dx ln cos x C ln sec x C 1 sec x dx ln sec x tan x C 2 cot x dx ln sin x C 3 csc x dx ln csc x cot x C 4 Integrating Powers of sin x and cos x sin m x dx cos n x dx Two strategies are employed when evaluating integrals of the form or m n where and are positive integers Try to discover the two strategies Evaluate the following integral cos5 x dx Evaluate the following integral sin 4 x dx or sin x are most easily evaluated by Strategies Integrals involving odd powers of cos x or sin x With even powers of cos x or sin x we use splitting off a single factor of cos x the power reducing half angle formulas Integrating Powers of sin x and cos x sin m x cos n x dx Different strategies are employed when evaluating integrals of the form where m and n are positive integers Try to discover the strategies Evaluate the following integral sin 4 x cos2 xdx Evaluate the following integral sin 3 x cos 2 x dx If you are given a combination of sines and cosines do one of the following 1 If the powers of sine and cosine are both even then rewrite both the sine and cosine as sin 2 x cos 2 x sin 2 x cos 2 x powers of and Next replace both and using the power reducing half angle formulas 2 If one of the powers of sine and cosine is odd then split off one multiple from the oddpowered function The function you split off will become part of du You should be able to rewrite the even powered portion that was leftover using a Pythagorean identity Next you should be able to combine the even powered functions The cofunction of the function you split off will become u Integrating Powers of sec x and tan x tan m x secn x dx Different strategies are employed when evaluating integrals of the form where m and n are positive integers Try to discover the strategies I will not allow the use of reduction formulas on the exams or quizzes You may use them when completing the online homework Evaluate the integral tan 6 x sec 4 x dx Evaluate the integral sec7 x tan 5 x dx Evaluate the integral sec x tan 2 x dx If you are given a combination of tangents and secants do one of the following sec2 x 1 Try splitting off a If you are left with an even number of secants then use a Pythagorean identity to rewrite these secants in terms of tangent What you split off will sec 2 x become part of du while u tan x If splitting off a left you with an odd number of secants then sec x tan x 2 Try splitting off a If you are left with an even number of tangents then use a Pythagorean identity to rewrite these tangents in terms of secant What you split off u sec x sec x tan x will become part of du while If splitting off a did not leave you with an even number of tangents and part 1 did not work then tan x sec x 3 Rewrite the even power of in terms of to produce a polynomial in sec x and integrate A reduction formula might be useful