3.7.5 Integration (A-level only)

A structural engineer models the rate of change of deflection $y\,$ of a loaded beam using the gradient function:

\frac{dy}{dx} = \frac{7x+1}{(x-2)(2x+1)^2}, \quad x > 2

Find the values of the constants $A$ , $B\,$ and $C\,$ such that

\frac{7x+1}{(x-2)(2x+1)^2} \equiv \frac{A}{x-2} + \frac{B}{2x+1} + \frac{C}{(2x+1)^2}

[4]

Hence find the exact change in deflection between $x = 3$ and $x = 4$ by calculating

\int_{3}^{4} \frac{7x+1}{(x-2)(2x+1)^2} \, \mathrm{d}x

giving your answer in the form $\ln q + r$ where $p$ , $q\,$ and $r\,$ are rational numbers.

[5]

3.7.5 Integration (A-level only) Questions

A structural engineer models the rate of change of deflection $y\,$ of a loaded beam using the gradient function:

\frac{dy}{dx} = \frac{7x+1}{(x-2)(2x+1)^2}, \quad x > 2

Find the values of the constants $A$ , $B\,$ and $C\,$ such that

\frac{7x+1}{(x-2)(2x+1)^2} \equiv \frac{A}{x-2} + \frac{B}{2x+1} + \frac{C}{(2x+1)^2}

[4]

Hence find the exact change in deflection between $x = 3$ and $x = 4$ by calculating

\int_{3}^{4} \frac{7x+1}{(x-2)(2x+1)^2} \, \mathrm{d}x

giving your answer in the form $\ln q + r$ where $p$ , $q\,$ and $r\,$ are rational numbers.

[5]