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First, recall how to construct a proof by simple induction in this manner: (1) Assume statement true for n=k, (2) Prove true for n=k+1, (3) Show true for n=1.(1) => ∑(from r=1 to k) r(2r−1)(3r−1)= (k/6...

Let's first find the equilibrium position of the spring. When the mass is first attached the spring will oscillate like a simple harmonic oscillator, in the real world the oscillator will eventually settl...

1/(5 + 4i) * (5 - 4i)/(5 - 4i) = (5 - 4i)/41

You need to use a clever trick for this! Write ln(x) as 1ln(x), and use integration by parts:u=ln(x) v'=1u'=1/x v=xThen applying the formula we obtain∫ln(x)dx = xln(x) - ∫[(1/x)x] dx = ...