Pearson Edexcel IAL Mathematics P3.5 Integration Question Bank

Practise algebraic integration of standard functions, recognised derivatives and definite integrals in exact forms and area contexts.

Exam points

integrate standard powers, trig and exponential functions before applying exact limits
use algebraic integration to produce forms involving ln, π or surds
find total area by splitting regions where a curve changes sign

Covered syllabus points

P3.5.1 - 1 P3.5.2 - Integration by recognition

[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

[Maximum number: 4]

Hence, using algebraic integration, find the exact value of

\int_{\frac{\pi}{12}}^{\frac{\pi}{8}}\left(5-4 \cos ^{2} 3 x\right) \mathrm{d} x

M1 A1

\int(3-2\cos 6x)\,dx=3x-\frac{\sin 6x}{3}

dM1 A1

\left[3x-\frac{\sin 6x}{3}\right]_{\pi/12}^{\pi/8}=\frac{\pi}{8}+\frac{2-\sqrt{2}}{6}

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Covered syllabus points

P3.5 - Integration question 1

P3.5 - Integration question 2