1
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So we want to take the indefinite integral of 4x^3 over x^4 plus 7 dx.

2
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So how can we tackle this? It seems like a hairy integral.

3
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Now the key inside here is to realize you have this expression x^4 + 7

4
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and you also have its derivative up here. The derivative of x^4 plus 7 is equal to 4x^3.

5
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Derivative of x^4 is 4x^3; derivative of 7 is just 0.

6
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So that's a big clue that u-substitution might be the tool of choice here.

7
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U-sub -- I'll just write u- -- I'll write the whole thing.

8
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U-Substitution could be the tool of choice. So given that, what would you want to set your u equal to?

9
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And I'll let you think about that 'cause it can figure out this part

10
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and the rest will just boil down to a fairly straightforward integral.

11
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Well, you want to set u be equal to the expression that you have its derivative laying around.

12
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So we could set u equal to x^4 plus 7. Now, what is du going to be equal to?

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du, I'm doing it in magenta. du, well it's just going to be the derivative of x^4 plus 7 with respect to x,

14
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so 4x^3 plus 0 times dx. I wrote it in differential form right over here,

15
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but it's a completely equivalent statement to saying that du, the derivative of u with respect to x,

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is equal to 4x^3 power. When someone writes du over dx, like this is really a notation to say

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the derivative of u with respect to x. It really isn't a fraction in a very formal way,

18
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but often times, you can kind of pseudo-manipulate them like fractions.

19
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So if you want to go from here to there, you can kind of pretend that you're multiplying both sides by dx.

20
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But these are equivalent statements and we want to get it in differential form

21
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in order to do proper use of u-substitution. And the reason why this is useful --

22
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and I'll just rewrite it up here so that it becomes very obvious;

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our original integral we can rewrite as 4x^3 dx over x^4 plus 7.

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And then it's pretty clear what's du and what's u.

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U, which we set to be equal to x^4 plus 7. And then du is equal to this.

26
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It's equal to 4x^3 dx. We saw it right over here. So we could rewrite this integral --

27
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I'll try to stay consistent with the colors -- as the indefinite integral,

28
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well we have in magenta right over here, that's du over -- try to stay true to the colors --

29
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over x^4 plus 7, which is just u. Or, we could rewrite this entire thing as

30
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the integral of 1 over u du. Well, what is the indefinite integral of 1 over u du?

31
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Well that's just going to be equal to the natural log of the absolute value --

32
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and we use the absolute value so it'll be defined even for negative use --

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and it actually does work out. I'll do another video where I'll show you it definitely does.

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The natural log of the absolute value of u and then we might have a constant there that was lost when we took the derivative.

35
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So that's essentially our answer in terms of u. But now we need to un-substitute the u.

36
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So what happens when we un-substitute the u?

37
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Well, then we are left with -- this is going to be equal to --

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the natural log of the absolute value of -- well, u is x^4 plus 7 -- not C, plus 7 --

39
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and then we can't forget our plus C out here. And we are done!