CAIE IGCSE Additional Math 14.11 Integrate Powers of X Questions

Practise integrating rational, logarithmic and algebraic expressions and reconstructing functions from derivative information in Cambridge Additional Mathematics problems.

Exam points

Integrate algebraic, logarithmic and rational expressions and recover a function from derivative information.

[Maximum number: 4]

Hence find $\displaystyle\int\dfrac{\ln3x}{x^{3}}\,\mathrm{d}x$ .

$\dfrac{\ln3x}{x^{2}}=\displaystyle\int\dfrac{1-2\ln3x}{x^{3}}\,\mathrm{d}x$ , so $2\displaystyle\int\dfrac{\ln3x}{x^{3}}\,\mathrm{d}x=\displaystyle\int\dfrac{1}{x^{3}}\,\mathrm{d}x-\dfrac{\ln3x}{x^{2}}$ oe. M1 for using integration as reverse of differentiation from part (a); allow their A,B, no extra terms.
$\displaystyle\int\dfrac{1}{x^{3}}\,\mathrm{d}x=-\dfrac{1}{2x^{2}}$ oe, nfww. B1, FT on their A.
$-\dfrac{1}{4x^{2}}-\dfrac{\ln3x}{2x^{2}}+c$ oe. B2; M1 dep for rearranging simplification and attempting to integrate $ax^{-3}$ . A1 must include +c.

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CAIE IGCSE Additional Math 14.11 Integrate Powers of X Questions question 1