Antiderivative of hairier expression Khan Academy English Czech

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Thai Tee Ponsukcharoen

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So our goal in this video is to take the antiderivative of this fairly crazy looking expression.

V tomto videu je naším cílem zintegrovat tento šíleně vypadající výraz.

Or another way of saying it is to find the indefinite integral of this crazy looking expression.

Neboli najít neurčitý integrál tohoto šíleně vypadajícího výrazu.

And the key realization right over here is that this expression is made up of a bunch of terms.

Hlavní je si uvědomit, že se tento výraz skládá z několika členů.

And the indefinite integral of the entire expression

A neurčitý integrál celého tohoto výrazu

is going to be equal to the indefinite integral of each

bude roven neurčitému integrálu jednotlivých členů.

Takže to bude rovno...

So this is going to be equal to, we could look at this term right over here, and just take the indefinite that,

Můžeme vzít tento člen a určit jeho neurčitý integrál.

7x to the third dx.

And then from that, we can subtract the indefinite integral of this thing.

A pak od toho odečteme neurčitý integrál tohoto členu.

So we could say this is, and then

Takže toto a pak minus

minus the indefinite integral of 5 times the square root of x dx.

neurčitý integrál 5 krát odmocniny z x dx.

And then we can look at this one right over here.

A pak se můžeme podívat na tento člen.

So then we could say plus the indefinite integral of 18 square root of x. Square roots of x, over x to the third dx.

Takže plus neurčitý integrál 18 odmocnin z x lomeno x³ dx.

And then finally, I'm running out of colors here,

A pak nakonec, už mi docházejí barvy.

finally I need more colors in my thing.

Nakonec... Potřebuju více barev.

We can take the antiderivative of this.

Integrál tohoto členu.

So plus the antiderivative of x to the negative 40th power dx.

Takže plus integrál x⁻⁴⁰ dx.

So I've just rewritten this and color-coded things.

Prostě jsem to přepsal a barevně odlišil.

So let's take the antiderivative of each of these. And you'll see that we'll be able to do it using our

Takže zintegrujme všechny členy a uvidíte, že nám na to bude stačit

whatever we want to call it. The inverse of the power rule, or the anti-power rule,

naše obrácené pravidlo derivace mocniny neboli pravidlo integrace mocniny,

whatever you might want to call it.

jakkoli tomu chcete říkat.

So let's look at the first one.

Začněme s tím prvním členem.

So we have-- what I'm going to do is, I'm just going to find the antiderivative without the constant, and just add the constant at the end.

Najdeme zatím integrál bez konstanty a konstantu přidáme až na konci.

For the sake of this one. Just to make sure we get the most general antiderivative.

A dostaneme tak ten nejobecnější integrál.

So here the exponent is a 3.

Takže tady je exponent 3. Můžeme ho zvýšit o 1.

So we can increase it by 1. So it's going to be x to the 4th.

Takže to bude x⁴.

Let me do that same purple color, or pink color.

Udělám to tou stejnou fialovou. Nebo růžovou.

It's going to be x to the 4th, or we're going to divide by x to the 4th.

Bude to x⁴. A děleno 4.

So it's x to the 4th over 4 is the antiderivative of x to the 3rd. And you just had this scaling quantity, the seven out front.

Takže x⁴ lomeno 4 je integrál x³ a tady jsme měli ještě koeficient.

So we can still just have the seven out front.

Těch 7 vepředu. Takže ještě přidáme 7 vepředu.

So we get 7x to the 4th over 4.

Takže máme 7x⁴ lomeno 4. V pořádku.

Fair enough. From that, we're going to subtract the antiderivative of this.

Od tohoto členu odečteme integrál tohoto.

Now at first this might not be obvious,

Teď možná není jasné,

that you could use our inverse power rule, or anti-power rule here.

že i na toto lze použít naše obrácené pravidlo derivace mocniny.

But then you just need to realize that 5 times the principal root of x

Ale stačí si jen uvědomit, že 5 krát odmocnina z x

is the same thing as 5 times x to the 1/2 power.

je to stejné jako... Je to stejné jako 5x na 1/2.

And so once again, the exponent here is 1/2. We can increment it by 1.

Takže zase stejně, exponent je 1/2, zvýšíme jej o 1.

So it's going to be x to the 3/2. And then divide by the incremented exponent.

Bude to x na 3/2. A pak děleno navýšeným exponentem.

So divide by 3/2. And of course we had this 5 out front,

Takže děleno 3/2 a ještě jsme měli vepředu 5,

so we still want to have the 5 out front.

takže ji tam budeme mít i teď.

Now this next expression looks even wackier. But once again, we can simplify a little bit.

Další výraz vypadá ještě šíleněji, ale opět to můžeme zjednodušit.

This is the same thing-- let me do it right over here--

Toto je stejné... Udělám to tady.

this is the same thing as 18 times x to the 1/2 times x to the negative 3 power.

Toto je stejné jako 18x na 1/2 krát x na -3.

x to the 3rd in the denominator is the same thing as x to the negative 3.

x na 3 ve jmenovateli je to stejné jako x na -3.

We have the same base, we could just add the exponents.

Máme stejný základ, jen sečteme exponenty,

So this is going to be equal to 18 times x to the 2 and 1/2

takže to bude rovno 18 krát x na 2 a 1/2.

Na 2 a 1/2, nebo ještě jinak,

Or another way of thinking about it, this is the same thing as 18 times x to the 5/2 power.

toto je stejné jako 18 krát x na 5/2.

Na 5/2. Udělal jsem to správně?

Did I do that right? Yeah. Negative 3. Oh sorry, this is negative 2 and 1/2.

Ano, -3... Pardon, tady bude -2 a 1/2

Let me make this very clear. And this is going to be the negative 5/2 power.

a tady bude -5/2.

x to the negative 3 is the same thing as x to the negative 6/2.

x na -3 je to stejné jako x na -6/2,

Negative 6/2 plus 1/2 is negative 5/2.

-6/2 plus 1/2 je -5/2.

So once again, we just have to increment this exponent.

Takže zase musíme zvýšit tento exponent.

So negative 5/2 plus 1 is going to be negative 3/2.

Takže -5/2 plus 1, to bude -3/2.

So you're going to have x to the negative 3/2. And then you divide by what your exponent

Takže máme x na -3/2 a pak děleno tím zvýšeným exponentem,

is when you increment it. So divide it by negative 3/2.

takže děleno -3/2.

And then you had the 18 out front.

A pak máme 18 vepředu.

And we obviously are going to have to simplify this.

A to budeme muset očividně ještě zjednodušit.

And then finally, our exponent in this term. Let me not use that purple anymore.

A nakonec exponent v tomto členu... Nebudu už používat tuto fialovou.

The exponent in this term right over here is negative 40.

Exponent v tomto členu je -40, když jej zvýšíme,

If we increment it, we get x to the negative 39 power, all of that over negative 39.

dostaneme x na -39, to vše lomeno -39.

And now we can add our constant, c.

A teď můžeme přidat naši konstantu c.

And all we need to do is simplify all of this craziness.

A teď už stačí jen zjednodušit tyto šílenosti.

So the first one is fairly simplified.

Ten první člen je už skoro zjednodušený,

We can write it as 7/4 x to the 4th.

můžeme jej napsat jako 7/4x⁴.

Now this term right over here is essentially negative 5 divided by 3/2.

U tohoto členu je -5 děleno 3/2,

So 5 over 3/2 is equal to 5 times 2/3, which is equal to 10 over 3.

takže 5 lomeno 3/2 je rovno 5 krát 2/3, což je rovno 10/3.

So this thing right over here simplifies to negative 10/3 x to the 3/2.

Takže tento člen se zjednoduší na -10/3x na 3/2.

And then we have all of this craziness.

A pak máme celou tuto šílenost.

Now 18 divided by negative 3/2 is

18 děleno -3/2.

equal to 18 times negative 2/3.

18 děleno -3/2 je rovno 18 krát -2/3,

Which is equal to, well we can simplify this a little bit,

což je rovno... Můžeme to trochu zjednodušit.

this is the same thing as 6 times negative 2. Which is equal to negative 12.

Toto je stejné jako 6... Toto je stejné jako 6 krát -2, což je rovno -12.

So this expression right over here is negative 12x to the negative 3/2.

Takže tento výraz je -12x na -3/2.

And then finally, this one right over here,

A konečně tento člen můžeme přepsat jako...

we can just rewrite it as, if we want, we could, well, we could just write negative 1/39 x to the negative 39 plus c.

Pokud chceme, můžeme prostě napsat -1/39x na -39. Plus c a máme hotovo.

And we're done. We've found the indefinite integral of all this craziness.

Našli jsme neurčitý integrál celé této šílenosti.

And I encourage you to take the derivative of this.

A doporučuju vám, abyste si toto zderivovali,

And you can do it using really just the power rule,

na což vám stačí pravidlo derivace mocniny.

to take the derivative of this.

Zderivujte si to a ověřte si,

And verify that it does indeed equal this expression that we took the antiderivative of.

že se to opravdu rovná výrazu, který jsme právě zintegrovali.

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