I want to talk about how we decide which integration technique we're going to use and I kind of want to do this with a a flowchart kind of a graphic organizer I'm not going to do a bunch of uh integration in this I'm going to maybe just talk about what these types of functions uh would look like and kind of my thought process on uh on deciding which technique to use so the first thing I would look for is this if it were a simp simple function that involved only addition and subtraction of a bunch of separate little pieces for example this you know it's x^2 7x 1 /x sin a bunch of just Little functions that I have a separate rule for and I could just easily do each individual piece and keep the addition between them and what I mean by that is you know 1/3 x cubed + 7 x^2 plus natural log of x X it's minus cine X put a plus c something like that where you just get to do each little piece if that's what your function is then you just proceed with integrating each little piece but if the answer is no next I look for if I instead had a product of two functions so two functions multiplied together I'm trying to integrate you know f ofx time G ofx so not a bunch of pieces uh added or subtracted but rather pieces that are multiplied together the first thing I would look for if it's a product is Will substitution work and what I look for with substitution is do I have a part like you can see this is x^2 * the S of X cubed if I were to make uux cubed we call it U substitution the derivative of that is 3x^2 and I could easily transform let me make that more obvious it's an X cubed I could easily transform you know the derivative into x^2 with a constant multiplier and uh I'm not going to go through the whole process just in the interest of time but again if you were uh X cubed the derivative of that would be 3x^2 and I could make a perfect match by doing this and I could integrate and it would basically it would turn my integral into 1/3 the integral of sin U du which is easily integrated now if substitution doesn't work and again that should be the first thing you check if that doesn't work you go to integration by parts and this is where again you could say well if U is X the derivative of that is one that does not help me get rid of the product If U was X2 I would end up with my du being inside my trig function that can't happen so you go to integration by parts and this is a you know a formula that you you use or it's this tabular integration method but essentially you pick one of your functions to be U and the other to be the derivative of V uh I can quickly do that right here you see I've just uh done my tabular integration and uh you could you can look at a different video of mine to kind of see that full process but that's what I would look for so for for products first check substitution if that works great if not good to buy parts what if it's not a product what if it's a quotient so my kind of standard way to think about quotients and again when we when we're talking about quotients we mean you know integrating a function divided by another function like this um the first thing I would check much like products is would substitution work and substitution would be a situation where you can see that you know making like the denominator your U and you say well my D my derivative of that is 2x so I could easily adjust like that and make it a perfect match and use substitution um that's what you're looking for uh could you be the denominator most of the time it'll be the denominator and then your derivative will help you get rid of the numerator that's kind of what to look for the next thing is um you know know uh does substitution not work then I would look for partial fractions partial fractions usually the denominator can be uh uh factored well always can be factored and then you can do the kind of you know a overx + 2 + B overx + 4 I'm not going to go through the whole process uh I have another video about how to do that but you would um use that method and again this is if substitution didn't work like here if you think about multiplying this out out you know x^2 + 6x + 8 if that was your U du would definitely not we wouldn't be able to do anything with that to to offset the three in the numerator so that's uh partial fractions now if partial fractions doesn't work and usually the reason would be well I I can't Factor the uh the denominator then I would look for trigonometric substitution so you see here x^2 + 4 you can use substitution cuz U would be 2X and there's no X up here you can't Factor this because it would require imaginary numbers x + 2 i x - 2 I so we would go to trigonometric substitution and look for like inverse tangent inverse sign now one last thing I just want to point out I'm just going to move this up a little bit um the other thing with quotients to kind of look for if the answer is no is uh can I do division okay and this is a kind of a rare thing and there's two different types to look for here one of those is the integral is something like this where the top is the more you know heavy part it's got more terms and the bottom is like a single term and you could say well this thing just reduces to 4 plus X you know if you just kind of divided each thing 4X / X and x^2 ID X and then you could just integrate that often times simplifying that just goes to to power rule so look for that if the denominator is super easy like a single term can you just separate out the numerator over that kind of common denominator the other thing to look for uh is potentially something like uh you know X2 over you know x^2 - 9 and here it's tempting to think you can go straight to partial fractions because you can Factor the problem is we have to do a pre poomi division on this because we have the same power on the top and bottom and really the idea behind um you know being able to do partial fractions is um we can you know Factor this out in a way that we can use kind of a a a modified substitution on these these factors well that's not going to work down here because there's no way X+ 3 and xus 3 in this denominator are ever going to be able to be modified in a way to get X squ in the numerator so you might have to use division like polom division um which again I have a video about you can go watch that but uh you also maybe could have a simpler division this to be honest is the most forgotten rule like this is the thing that people most often miss um so just be be aware of it now real quick if you are a cal AB student you are only responsible for these kind of like substitutions and maybe like this kind of simple division okay if you're a BC student then you're responsible for all of them so kind of uh uh study that uh you know that flowchart that's kind of the way I would I would be thinking about uh these things is uh you know is it a simple thing if it's a product check substitution if that doesn't work do by Parts if it's a quotient you had a lot of things to check substitution first and then maybe partial fractions trig substitution if this video helped you please like And subscribe for more math help