Hello and welcome to My Secret Math Tutor,

in this video we are going to cover some of the basic rules for indefinite integrals. Now before we get too far there is one big

important thing you really need to point out, and that is know the difference between the

types of integrals that are out there. In this video we are going to specifically

cover indefinite integrals, but there are integrals called definite integrals, and I’ll

be covering those in a different video. The biggest difference you can really spot

with these, is that you’ll see numbers right next to the integral symbol. So those are handled differently, we think

of those as an area, but again, we’ll cover those in a different video, let’s just focus

on the basics, the indefinite integrals. Now when it comes to these, the good news

is that there is only a few simple rules you really have to keep in mind. And probably the biggest rule that you want

to keep in mind is that with these indefinite integrals you’re really thinking of an anti-derivative. So you’re trying to work backwards to figure

out you know, if I was to take the derivative of this function, what would I get. And often times, you’ll see notation for an

anti-derivative simply use a capital letter. So what this is trying say is that I’m starting

with some sort of function, and I’m working backwards and its anti-derivative is now over

here, which I’m calling capital F. These anti-derivatives will also involve a plus C, or constant. And that’s because when we are taking the

anti-derivative, it may have some sort of constant component, but usually the derivative

wipes that out, so we’re not even sure what that is. Now this is probably the biggest thing you

have to keep in mind, but there are a few other things that are good about anti-derivatives,

or indefinite integrals, and that is they have a sum and difference rule, so if you

have two functions that are added or subtracted, then you are simply taking the anti-derivative

of each of those and adding or subtracting respectively. If you have a constant multiplied out front,

feel free to pull out that constant, then you can just worry taking the anti-derivative

of the function that is left on the inside. Alright, so there is not a whole lot of rules,

and that’s kinda a nice thing, so there is not a lot to remember, lets go ahead and jump into

some examples and see how they play out. So for the first of these I’m going to look

at the indefinite integral of 3x^4. And I could also say I’m simply looking for

the anti-derivative of 3x^4. Alright, so the biggest thing you want to

think of is, if I was to start with a function and take its derivative, then it would turn

into this. And to get really good at that process you

have to start really knowing your derivatives. So for example I might start with playing

around saying when I take derivatives I bring down the power and reduce it by 1, so if I

was doing that I think that this power must have been 5, you know that must be the only

way I could reduce it and it would end up as 4. So in going in the other direction I would

think of adding 1 to the power. Alright, so the 3 is a constant, so I’m not

going to worry about that, in fact its just going to hang out for a bit. Um and lets see. What should we put out front. You know I’m going to say that that is a 1/5,

and we’ll put our constant. So its a really big process to think backwards

like that, but let’s go ahead and check. If I was to take the derivative of this, you

know derivatives they also are not effected by these constants out front. But I would bring down the power, it would

cancel out the 5, and this would be reduced to by 1 turning it into 4. So sure enough we see that our anti-derivative

is 3/5 x^5 + C. And this what we would have for the indefinite integral. Alright, lets try this again, with another

example. This one I’m going to try and walk through

a little bit more, just so we can see what’s going on here. So if you see a constant and it is not being

multiplied by anything, you can consider it to be a variable with a power of 0. So here is a cos(x) dx. And again we want to think backwards. So I’m thinking about this power, let’s see. We’ll add 1 to the power. So 0 + 1 is 1. And then I’m going to end up dividing by this

new power, so 4 divided by 1 is simply 4. So that looks pretty good. Moving on to cosine, what function’s derivative

would equal cosine, well that would have to be sine. And of course we put a plus C on the end. I could clean this up a little bit, and just

say that this is 4x – sin(x) + C. And that would be my anti-derivative. So you’ll notice that one thing that can make

this a much easier process is knowing your derivatives really really well. The more derivatives rules you have in back

pocket, the better you’ll recognize your anti-derivatives and be able to move in that direction. One of the biggest though is probably this

power rule for derivatives, and they way you are going to see it show up for anti-derivatives

is when you have something like x to a power you are going to add one to the power, and

then divide by that new power. So its exactly like the usual power rule,

but its doing opposite things, its adding 1 to the power its dividing by the new power,

and of course you saw me use that in a couple of the last examples. The more of these derivatives rules you know,

again the better this will be. So know your trigonometric rules, know your

inverse trigonometric rules, know all of those. Alright let’s get into another example and

see some things that can make this go a little bit better. Now with this one, um you have to be really

careful how you apply anti-derivatives. How you work backwards because I have something

divided by something else, but if you think all the way back to those rules for indefinite

integrals, they said nothing about division. So for problems like this sometimes its best

to actually rewrite them. In some of my derivative videos, this was

a handy technique. If you just change what it looks like, sometimes

you can handle it fairly quickly, and use the rules that you actually have. So so far I’ve just split this up into two

fractions, looks a little strange, but it is a valid rule. You know if we were to put these back together

I’d make sure they have the same denominator, and sure enough we’d end up back over here. Now let’s see, moving on I might look at this

in stead of a square root, I would say this is like x^(1/2). What I’m going to end up doing here, is that

I want to reduce these powers and so let’s see. If I want to reduce this first power, I’d

subtract the 1/2 from the 2. So lets see, 2 minus 1/2 I’d be left with

3/2. And here since there is no x’s in the top

I can just consider this as a negative exponent so x^(-1/2). So notice we have not taken the anti-derivative. I haven’t even thought about that process

yet. I’ve really just gone through a process of

manipulating this so that I can start using some more of my rules, and now we can actually

use that rule of adding 1 to the power, and then dividing by the new power. So let’s go ahead and do that. So I’m going to take my x, we’re going to

take the power, what it is now, add 1 to the power, and what ever new number this is, we’re

going to divide by it. So of course we have some work to do, that’s

what’s going to end up over there, and we have to do the same thing for this so -1/2

+ 1 and we are going to divide this by what ever new number we get. So that looks pretty good. We should probably put a plus C in there,

and let’s go ahead and clean this up. So 3/2 + 1 that would equal x^(5/2). So that’s going to be this number down here. 1 divided by 5/2. And let’s see x to the 1/2, so -1/2 + 1=1/2. And that’s this number down here. It looks a little messy and of course we should

probably clean this up, when we divide by a fraction we want to flip and multiply. So let’s go ahead think what this is really

doing, so this is 1 divided by 5/2, or we can think of it as 1 multiplied by 2/5. So this expression becomes 2/5 multiplied

by x^(5/2) plus we’ll do the same thing here, this will be a 2x^(1/2), and we have our plus

constant. Ok, so I had to really manipulate this thing

so that I was only using the anti-derivative of things that were added together, and in

this case just using a power rule to add 1 to the power, and reduce it. Now be very very careful when you are doing

anti-derivatives. These are the most common mistakes someone

will do. Sometimes you have two functions that are

multiplied together, and people will try and try and take the indefinite integral of each

of those, this is not a valid rule, in fact you are going to handle that later, using

something, a different rule for these anti-derivatives, you’ll get into u-substitution, and stuff

like that. Another one is what looks like a chain rule,

if you have a function inside of another one, people will try and take the anti-derivative

of the outside, and the anti-derivative of the inside, but again that is not a valid

rule, you’ll usually take care of those using something like u substitution. Ok. And of course the biggest, the biggest rule

that you, or not necessarily rule, but the biggest problem that some people have with

anti-derivatives is even a very simple function might not have a closed form for the anti-derivative. That means when I try and write down the formula

or just write down what the anti-derivative is, its not going to be simple as you think

it is. Its not going to be a single finite formula

that describes whats going on. So this happens to be one of those ones that

does not have a closed form, even though it looks really simple you’ll probably have to

wait until a little bit later in calculus to see what this guys anti-derivative is,

and its going to involve an infinite amount of functions, so be careful for those. Alright let’s get back to some ones that we

can do, or at least ones that you know it looks like maybe we can’t do them, but again

if we manipulate them just right we have all the tools we can actually take an anti-derivative. Alright, let’s try this guy out. So here I have (x^4 -1)^2, and some common

mistakes might be to try and add 1 to this power and divide by 3, and continue on from

there. But we really have another function inside

of that, ok, so we have to be a little bit more careful, we want to think of manipulating

this first. I’m just going to spread this out. So this is (x^4 -1) multiplied by (x^4 – 1). And I’m doing this so that I can expand it

out into a whole bunch of functions that are added together. Then we’ll go ahead and take care of those. Alright, so let’s keep expanding. x^4 times x^4 we’ll add those together those

exponents, outside we’ll be minus x^4, inside minus x^4, last terms minus 1 times minus

1, plus 1. And let’s go ahead and combine the common

terms from here. So x^8 minus 2x^4 plus 1, ok that’s all good,

now that its all spread out, now I can worry about actually taking this anti-derivative. And of course we’ll just use a power rule. So starting with the first one 8 + 1 is 9,

we’ll divide by that new power. 4+1 is 5, we’ll divide by that new power. The 2 will still be there. And this one it looks like I don’t have an

x, so we’ll just consider this as x^0. We want to add 1 to that power, so we’ll get

x^1 and just to make sure we are on the same page, let’s put a plus C, there is the proper

indefinite integral. Alright, now as a big challenge to this we

are going to do just one more example, but his last one really shows you why its important

to know all of your derivatives. This one you know when you are first doing

integrals looks pretty tricky because we don’t have any rules to take care of division, and

we only have so many limited things to do for trigonometric functions, but check this

out. This first expression is really a special

type of derivative. In fact it’s the derivative for inverse tangent. And over here this is guy is the derivative

of just regular old tangent. So in other words what I’m trying to say here

is that if I took this guys derivative I would get this, if I took tangent’s derivative I

would get secant squared, and since anti-derivatives, this indefinite integral, works backwards

these are the two expressions I would get. So I would get arctangent plus tangent, and

of course our plus constant. So even though it looks pretty scary looking,

and it looks like we might not have any rules to take care of it, its all about knowing

your derivatives and you’ll be just fine. Alright, thanks for watching this video, please

check out some of my other calculus videos, I’ll be doing more about integration in just

a little bit, but hopefully this gets you started with indefinite integrals. Thank you for watching.