a) (1.5 points) In a laboratory experiment, 5 measurements of the same object have been made, which gave the following results: $\mathrm { m } _ { 1 } = 0.92 ; \mathrm { m } _ { 2 } = 0.94 ; \mathrm { m } _ { 3 } = 0.89 ; \mathrm { m } _ { 4 } = 0.90$; $\mathrm { m } _ { 5 } = 0.91$. The result will be taken as the value of x such that the sum of the squares of the errors is minimized. That is, the value for which the function $E ( x ) = \left( x - m _ { 1 } \right) ^ { 2 } + \left( x - m _ { 2 } \right) ^ { 2 } + \left( x - m _ { 3 } \right) ^ { 2 } + \left( x - m _ { 4 } \right) ^ { 2 } + \left( x - m _ { 5 } \right) ^ { 2 }$ reaches its minimum. Calculate this value x . b) (1 point) Apply the integration by parts method to calculate the integral $\int _ { 1 } ^ { 2 } x ^ { 2 } \ln ( x ) d x$, where ln denotes the natural logarithm.
a) (1.5 points) In a laboratory experiment, 5 measurements of the same object have been made, which gave the following results: $\mathrm { m } _ { 1 } = 0.92 ; \mathrm { m } _ { 2 } = 0.94 ; \mathrm { m } _ { 3 } = 0.89 ; \mathrm { m } _ { 4 } = 0.90$; $\mathrm { m } _ { 5 } = 0.91$.
The result will be taken as the value of x such that the sum of the squares of the errors is minimized. That is, the value for which the function
$E ( x ) = \left( x - m _ { 1 } \right) ^ { 2 } + \left( x - m _ { 2 } \right) ^ { 2 } + \left( x - m _ { 3 } \right) ^ { 2 } + \left( x - m _ { 4 } \right) ^ { 2 } + \left( x - m _ { 5 } \right) ^ { 2 }$ reaches its minimum.
Calculate this value x .
b) (1 point) Apply the integration by parts method to calculate the integral $\int _ { 1 } ^ { 2 } x ^ { 2 } \ln ( x ) d x$, where ln denotes the natural logarithm.