2025 MI P1 Q5

Using the substitution \(u = x^2 + 3\), show that\[\displaystyle \displaystyle\int \frac{x^3}{\sqrt{x^2 + 3}} \, \mathrm{d}x = \frac{1}{3}\sqrt{(x^2 + 3)^3} - 3\sqrt{x^2 + 3} + C.\][4]

Hence find the exact value of \(\displaystyle \displaystyle\int_{-1}^{3} \left| \frac{3x^3}{\sqrt{x^2 + 3}} \right| \, \mathrm{d}x\), leaving your answer in the form \(a + b\sqrt{3}\) where \(a\) and \(b\) are integers to be determined.[3]