Calculus – The basics of indefinite integrals


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.

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