A geologist models the rate of thermal energy emission, $P$ , from a cooling volcanic vent using the equation

\frac{8 + 2t^3 + \ln t}{t^4}, \quad t \ge 1

where $P\,$ is measured in megawatts and $t\,$ is the time in hours since initial observation.

Find $dt\displaystyle \int \frac{\ln t}{t^4} \,\text{d}t$

The total energy $E$ , in megawatt-hours, emitted by the vent between $t = 1$ and $t = 2$ is given by the area under the curve $P (t)$ between these two times.

Use your result from part (a) to calculate the exact value of $E$ . Give your answer in the form $\ln b$ , where $a\,$ and $b\,$ are constants to be determined.

A geologist models the rate of thermal energy emission, $P$ , from a cooling volcanic vent using the equation

\frac{8 + 2t^3 + \ln t}{t^4}, \quad t \ge 1

where $P\,$ is measured in megawatts and $t\,$ is the time in hours since initial observation.

Find $dt\displaystyle \int \frac{\ln t}{t^4} \,\text{d}t$

The total energy $E$ , in megawatt-hours, emitted by the vent between $t = 1$ and $t = 2$ is given by the area under the curve $P (t)$ between these two times.

Use your result from part (a) to calculate the exact value of $E$ . Give your answer in the form $\ln b$ , where $a\,$ and $b\,$ are constants to be determined.

1.11.5 Integration by substitution and by parts (A-level only)

Question 59

Markscheme

1.11.5 Integration by substitution and by parts (A-level only)

Question 59

Markscheme

1.11.5 Integration by substitution and by parts (A-level only)

Question 59

1.11.5 Integration by substitution and by parts (A-level only) Questions

1.11.5 Integration by substitution and by parts (A-level only)

Question 59

1.11.5 Integration by substitution and by parts (A-level only) Questions