2021HSCShort answer

Q12 (a)

3 marks

Question

Find $\int\frac{2x+3}{x^2+2x+2}\,dx$ .

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Solution

Criteria	Marks
Provides correct solution	3
Integrates one of the correct fractions OR writes the integrand as the correct sum of fractions and completes the square in a denominator, or equivalent merit	2
Attempts to write the integrand as a sum of two suitable fractions, or equivalent merit	1

Sample answer:

$\begin{aligned}\int\frac{2x+3}{x^2+2x+2}\,dx=\int\frac{2x+2}{x^2+2x+2}\,dx+\int\frac 1{x^2+2x+2}\,dx\\&=\ln(x^2+2x+2)+\int\frac1{1+(x+1)^2}\,dx+C\\&=\ln(x^2+2x+2)+\tan^{-1}(x+1)+k\end{aligned}$

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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