Can a math genius tell me how to do this?

Well, Wolfram|Alpha couldn’t solve the problem…

Here’s the link. It won’t format correctly. http://www.wolframalpha.com/input/?i=integrate+1/(8x%E2%88%9A(x(1%2B%E2%88%9Ax)(%E2%88%9A(1%2B%E2%88%9Ax)-1)

Oh, and I can’t tell whether that says 8x or 8 times…

1) you need to find the square route of x
2) find the square route of x + 1
3) subtract 1 make a note of what you got
4) square route of x + 1 times x make a note of what you got
5) multiply the 2 values you made a note of together
6) find the square route of that
7) multiply by 8
square route of x is x to the second power or xx
I’m not 100% sure but I’m pretty sure.

1) Distrute the square root
1/(8*sqr(x)*sqr(1+sqr(x))*sqr(1+sqr(x)))
2) Eliminate the sqr(1+sqr(x))‘s
1/(8*sqr(x)*(1+sqr(x))
3) Distribute
1/(8*(sqr(x)+sqr(x)*sqr(x))
4) Eliminate sqr(x)‘s
1/(8*(sqr(x)+x)
5) Distribute
1/(8*sqr(x)+8*x)

Hope you can take it from here.

@daytonamisticrip Square root of x is not x to the second power. For example, the square root of 4 is 2, while 4 to the second power is 16. Also, there is no value of x. The question is not to plug in a value for x and simplify the expression, the question is to integrate the function, thereby ending up with another function.

@bigjay I’m sorry I can’t help you right now. I looked at the problem, but nothing jumped out at me as a good way to approach it. Unfortunately, I don’t have the time to sit and mess with it. I’d really love to, because it looks like an interesting question. I will say that my first thought was trig sub, but I’m not sure if it will get you anywhere. The other thing I thought of was to see if you can simplify the expression and go from there.

EDIT – It looks like @PhiNotPi might have beaten me to the second idea.

The square root of X is the same as X^0.5

If it’s as written in the attached png, then I don’t think PhiNotPi’s solution works, sadly, because of where the square root signs are. Are you sure it’s written correctly there? If it doesn’t simplify, then it looks to be pretty crazy hard to solve.

Clarification, please—What does the first ‘X’ represent in the expression:
dy/dx = 1/(8×√(x(1+√x)(√((1+√x)
Does it mean multiply, or does it represent the variable x?

One technique that might work is to make a change of variable.
Let u = 1+√x
Then x = (u-1)^2

The big radical sign in the denominator then reduces to
(u-1)^(3/2) u^(½)

So the integrand becomes (assuming “times” rather than variable x):
1/8 (u-1)^(-3/2) u^(-½)
Not that I can integrate that, either… It just looks simpler. ~