A chemical engineer models the rate of heat loss H(T)H(T)H(T), in kilowatts per degree Celsius, from a specific pressurized vessel using the formula
H(T)=2T2−3T−7T−4,T>4 H(T) = \frac{2T^2 - 3T - 7}{T - 4}, \quad T > 4 H(T)=T−42T2−3T−7,T>4where TTT is the temperature of the vessel in degrees Celsius.
Write H(T)H(T)H(T) in the form
AT+B+CT−4 AT + B + \frac{C}{T - 4} AT+B+T−4Cwhere AAA, BBB, and CCC are integers to be found.
Hence use algebraic integration to show that the total heat loss between temperature T=5T = 5T=5 and T=7T = 7T=7, given by ∫57H(T) dT\int_{5}^{7} H(T) \, dT∫57H(T)dT, is equal to
α+βln3 \alpha + \beta \ln 3 α+βln3where α\alphaα and β\betaβ are integers to be found.
