1.9.27 Integration by substitution (reverse chain rule) (A-level only)

A scientist models the rate of temperature change, $dTdt\frac{dT}{dt}$ , of a new alloy using the equation

\frac{dT}{dt} = \frac{9t}{3t^2 + k}

where $\ge 0$ is the time in seconds and $k$ is a positive constant.

Find

\int \frac{9t}{3t^2 + k} \, dt

[2]

Given that the change in temperature between $t = 0$ and $t = 2$ is exactly $ln⁡125\ln 125$ degrees, determine the value of $k$ .

[4]

1.9.27 Integration by substitution (reverse chain rule) (A-level only) Questions

A scientist models the rate of temperature change, $dTdt\frac{dT}{dt}$ , of a new alloy using the equation

\frac{dT}{dt} = \frac{9t}{3t^2 + k}

where $\ge 0$ is the time in seconds and $k$ is a positive constant.

Find

\int \frac{9t}{3t^2 + k} \, dt

[2]

Given that the change in temperature between $t = 0$ and $t = 2$ is exactly $ln⁡125\ln 125$ degrees, determine the value of $k$ .

[4]