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

[XTTX]

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

Homework Equations

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

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.
\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]

 

[tim_lou]

you probably differentiate the wrong part of the integrand when using tabular methods.
try to break up

x^{999} dx
\ln(x)

 

[XTTX]

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!

 

[tim_lou]

so, you use ln(x) as the differential portion. what does integration by part tell you?

 

[XTTX]

I'm not sure I follow, but you mean:
\int udv = uv-\int vdu

 

[tim_lou]

so, yes, and? (edit: you don't need to use tabular method as using integration by part once is enough!)

 

[XTTX]

Alright, thanks. I got it now (I believe)
I just didn't know that \int ln(x)=xln(x)-x

Edit: also, just to confirm, is this the correct solution?
\frac{x^{1000}ln\left(x\right)-1.999x^{1000}}{1000}

 

[tim_lou]

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)?