Edexcel A-Level Mathematics A2 P3.5.1.1 Questions

Practise standard-function integration, exact definite integrals and using velocity or area integrals to form equations and totals.

Exam points

Integrate standard trig, exponential or power terms and keep constants of integration where needed
Evaluate definite integrals with exact bounds to give answers involving ln, π or surds
Use an integral of velocity or area to form an equation or exact total area

[Maximum number: 4]

g(x)=\frac{2 x^{2}-5 x+8}{x-2}

Hence use algebraic integration to show that

\int_{4}^{8} \mathrm{~g}(x) \mathrm{d} x=\alpha+\beta \ln 3

where $\alpha$ and $\beta$ are integers to be found.

$\int\left(2x-1+\frac{6}{x-2}\right)\,dx=x^2-x+6\ln(x-2)$ $\int_4^8 g(x)\,dx=\left[x^2-x+6\ln(x-2)\right]_4^8$ $=56+6\ln6-12-6\ln2$ $=44+6\ln3$
Therefore $\alpha=44$ , $\beta=6$ .

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Edexcel A-Level Mathematics A2 P3.5.1.1 Questions question 1