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.