Integration using trigonometric identities[list=1][*]Trigonometric integrals and[/*][*]Integration by substitution of trig. functions (or "trig. substitution")[/*][/list]Here are the main identities to keep in mind when doing integrals of trigonometric functions:[br][br][br][img width=254,height=273]https://lh6.googleusercontent.com/lykWAdZxGhNGCZSF1naqDR5mgAEQft7EvguKn3SCdjv0pDwTjxr1x8E8J2NJyg2Pu1EswkFaU0wxOfx_8rAef2udUcZCkYnS3aQxypITJ9badTkfS4wSj48Zv5Rq-CGf08cCYNNb[/img][br][br][br]Now we'll use them to solve integrals of the type[img width=129,height=36]https://lh5.googleusercontent.com/niVFvYuFP0yvEUIdTczjmE-eHwyEI0UmJCbksPqxjOR7JT3eJMp7pAtqPLOdKYyHp43OXmYBvWnoGUeeGdAlvHb1UMkjLIqK7a2rPEVtCoxs_LEosHLn4_AhgWDpogdV5YMv_3wz[/img][br][br]These will be divided into two categories below, those where at least one of m or n are odd, and integrals where both m or n are even. The best way to do this is by example, so here we go. Example: ∫ tan3x dxFind the indefinite integral:    [br][br]This is an example very similar to the first, but it will be a little more complicated to integrate. [br][br]First, use the Pythagorean identity for tangent and secant[br][br][br][img width=365,height=61]https://lh6.googleusercontent.com/frxZb1Hdseu_jofD1I0x2rG5OA4N1gj-kkZqNRKqzy2QkECwSlVGt5YRktb01QHCXsZ1pEonbuhJnBs7tI1HH6LjOgITErYU--6TBKkfKapkd2vzdkJdyc0YcgM2_9B5BOaJhwZs[/img][br][br][br]From this, we get two integrals, each of which can be solved by simple substitution (u-substitution). Here's what it looks like:[br][br][br][br][img width=487,height=250]https://lh3.googleusercontent.com/KlqaExZYdHxVdEUa-fiy6I7nPbQXEm345r4isMzOWNtGHa6fRw4X2QV44xihKqOYJroUK99syBDoLli0iDdVu97yWmLc3h3mmRvKg3Ebn5kVJDCK9PL3tb0dfx3NYWwKfm4jDTcU[/img][br][br][br]Putting it all together (and remembering that the integral on the right had a minus sign in front of it), we get the result:[img width=293,height=44]https://lh5.googleusercontent.com/5voh6zta7wY3egeoPi5zjAQVt7ZU3W1RmW9FFa2-HdoG5SPweUjzpYQ-l68oal1zZABfH_IRv8u4BfuHpxEqRwQU3YwLhj2u0y862fPa6fu2tAeOhyZsWk2vq704eO4R_APrsMv2[/img] [br][br][br][br]