Integrate x^999 ln(x) - Find the Answer

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In summary, the conversation discusses finding the integral of [x^999 ln(x)] using integration by parts. The individual first attempts to use a tabular analysis method, but then realizes that it is not necessary. They then clarify the use of ln(x) as the differentiable part and discuss the correct formula for integration by parts. Finally, they confirm the correct solution and discuss the derivative of ln(x).
  • #1
XTTX
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Note: I already found "an answer," and as far as I can tell it's correct; however, once you see it, you'll know why I'm suspicious of it.

Homework Statement


Find [tex]\int\left[x\stackrel{999}{}ln\left(x\right)\right]dx[/tex]

Homework Equations


[tex]\int\left[x\stackrel{999}{}ln\left(x\right)\right]dx[/tex]

The Attempt at a Solution


I first started by making a tabular analysis because I believed that it would be necessary to use integration by parts. Where u = ln(x) and dv = x^999dx. Here's the expression I came up with, if you can even call it an expression. As you can see, it requires the use of sigma, which I've never had to use in this class thus far. Keep in mind that this is a high school Calculus BC class, so nothing too difficult.
[tex]\frac{x\stackrel{1000}{}}{1000}\left[ln\left(x\right)-\sum^{\infty}_{n=1}\left[\left(n-1\right)!\frac{1000!}{\left(1000+n\right)!}\right]\right][/tex]
 
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  • #2
you probably differentiate the wrong part of the integrand when using tabular methods.
try to break up

[tex]x^{999} dx[/tex]
[tex]\ln(x)[/tex]
 
  • #3
Well, we never learned the integral of natural log, so I had to use ln(x) as the differentiable portion of the equation.

And what do you mean by "break up?"

Thanks!
 
  • #4
so, you use ln(x) as the differential portion. what does integration by part tell you?
 
  • #5
I'm not sure I follow, but you mean:
[tex]\int udv = uv-\int vdu[/tex]
 
  • #6
so, yes, and? (edit: you don't need to use tabular method as using integration by part once is enough!)
 
Last edited:
  • #7
Alright, thanks. I got it now (I believe)
I just didn't know that [tex]\int ln(x)=xln(x)-x[/tex]

Edit: also, just to confirm, is this the correct solution?
[tex]\frac{x^{1000}ln\left(x\right)-1.999x^{1000}}{1000}[/tex]
 
Last edited:
  • #8
the second x^1000 seems to have the coefficient wrong.

you don't need to use the integral of ln(x) at all, since like you said, you are differentiating ln(x)=u, and integrating the other part, dv.

Apply the integral by parts formula, write out everything step by step.
dv=x^999 dx, u=ln(x)
what would you get?

Also, what is the derivative of ln(x)?
 

1. What is the formula for integrating x^999 ln(x)?

The formula for integrating x^999 ln(x) is ∫x^999 ln(x) dx = (x^1000/1000) (ln(x) - 1) + C.

2. Can x^999 ln(x) be integrated using basic integration techniques?

No, x^999 ln(x) cannot be integrated using basic integration techniques. It requires the application of integration by parts.

3. What is the value of the constant C in the integrated form of x^999 ln(x)?

The value of the constant C cannot be determined without additional information, such as the limits of integration.

4. Can the integration of x^999 ln(x) be simplified further?

Yes, the integration of x^999 ln(x) can be simplified further by using the properties of logarithms, such as the power rule and the product rule.

5. How can the integral of x^999 ln(x) be used in real-world applications?

The integral of x^999 ln(x) can be used in various fields of science and engineering, such as in calculating the work done by a variable force or in finding the average value of a function over a given interval.

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