Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x\sqrt{x+1}\right)=\dfrac{3x+2}{2\sqrt{x+1}}\).[2]
The diagram shows part of the curve \(y=\dfrac{3x+2}{\sqrt{x+1}}\). The curve intersects with the line \(2y+3x=20\) at point \(P\). The \(x\)-coordinate of the point \(P\) is 3.
Find the area of the shaded region bounded by the curve, the line and the \(x\)-axis.[8]
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