Pearson Edexcel IAL Mathematics P3.5.1 1

Practise integrating standard functions such as sin(kx), e^kx and sums of terms, then applying limits in exact answers.

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 $\alpha=44$ , $\beta=6$ .

M1 A1ft dM1 A1

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