Trigonometric Substitution There are forms depending on where u2 and a2 are These forms utilize trig ID formulas Best to set your ?a? and ?u? equal to something early sometimes not just simple variables consider the roots Sometimes u and a will need coefficients like ?2? and ?3? if they are numbers that need to be square rooted to fit the formula If both ?a? and ?u? have coefficients use root coeff ?a?/root coeff ?u? Remember to take the derivative of what you set x= Multiply this to the integral If you have a leftover ?x? or ?x2? go back to how you defined x= and just plug in You have what you need to plug in, it is just something you did at the beginning Simplify once you have all components of integral in right form Manipulate using trig ratios Use trig IDs to simply to a expression you know how to integrate (this is more prevalent option) Integrate You are not finished yet because you need to replace the angles with variables There should be no angles in the final answer Refer back to the x= at the beginning Make a reference triangle and find the trig ratio you need and then plug in Completing the Square Basic procedure for completing the square Use only the components that have x Put constant off to the side Take half the ?x? term, square it, add it to the x2 and x term Leave this part in parenthesis ALSO subtract your outside constant from this number Be wary to distribute whatever is outside the parenthesis before you take new number out and introduce it to the constant Maybe negative Maybe multiplied by a factor You can use the same ?a? ?u? substitution rules if you need a 1 to adhere to the trig IDs Usually involve trig IDs once you have found the squares Left over x values- you HAVE WHAT YOU NEED to substitute for these values Don?t forget to multiply by the derivative and simplify where you can Use trig IDs/ manipulation to change the integrand into something manageable Expression doesn?t need to have 3 parts to be completed As long as it has ?x? and ?x2? Gypsy magic that involves another component in addition to completing the square When you have polynomial on top and bottom (lower power on top) not just 1 on top Do something that resembles partial fractions Take deriv of the bottom polynomial like the top Set numerator = A(deriv)+B Then set Ax terms=x terms B terms and A terms= constants Solve for A &B Then split the integral into 2 (add them together) the A and B values will go out side integral sign since they are just constants Same denominator One has derive of denom as top (A) One has 1 as top (B) Deriv as top solve using u sub The deriv of the bottom is already in there so it works out perfectly Ln (bottom) keep A in front 1 as top NOW complete the square Use inverse trig integrals keep B in front