How Does the Substitution Rule Work?

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Homework Statement


If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Integral f(g(x)g'(x)dx=Integral f(u)du.


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The Attempt at a Solution

 
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Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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