Given that \(y=x^3\ln x\), find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\).[2]
Hence find \(\displaystyle\int_1^2 3x^2\ln x\,\mathrm{d}x\), giving your answer in the form \(\ln a+b\), where \(a\) is an integer and \(b\) is a rational number.[4]
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