With origin $O$, the curve $y={{x}^{2}}\ln x$ and the line $y=-x+2+4\ln 2$ for $x\ge 1$

Question

With origin $O$, the curve $y={{x}^{2}}\ln x$ and the line $y=-x+2+4\ln 2$ for $x\ge 1$ meet at the point $P$. Verify that the coordinates of $P$ are $\left( 2,4\ln 2 \right)$.[1] Show that the area of the shaded region bounded by the curve, the line and the $x$-axis is $a+b\ln 2+c{{\left( \ln 2 \right)}^{2}}$, where $a$, $b$ and $c$ are constants to be determined. [4]

Accepted Answer

$P=(2,4\ln2)$ $a=-\dfrac79$, $b=\dfrac83$, $c=8$

2023 VJC P1 Q1

2023 VJC P1 Q1

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