All it gives me is a pretty restatement of my original request. When I use WolframAlpha PRO, I get a result of 624510 and a visual representation of the area in question. I know the ends of the interval [0,pi] are problematic. What am I doing wrong? The 'definite integral' in WolframAlpha PRO was given x^3*ln(2*sin(x))^8 in the 'function to integrate' box. Ideally, what would be best, is if both programs would give me a closed form expression, if it exists. NIntegrate was not helpful either. You should know I am very new to Mathematica.

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    $\begingroup$ Partly because square brackets are reserved in Mathematica for containing arguments to functions. Parentheses () are the only symbols used in Mathematica for grouping terms and factors in mathematical expressions. That said, I ran that integral in my copy of Mathematica and it returned unevaluated, suggesting that it doesn't know an analytic form (which doesn't mean it doesn't exist; perhaps the expression could be massaged into a form that can be integrated analytically). $\endgroup$ – march Mar 29 '16 at 18:53

There doesn't look to be an analytic form of the integral (as mentioned in the comment by march), though numeric integration does match the expected result.

NIntegrate[x^3*Log[2*Sin[x]]^8, {x, 0, π}]

You can plot the corresponding area with

Plot[x^3*Log[2*Sin[x]]^8, {x, 0, π}, Filling -> Axis]

enter image description here

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