2023HSCShort answer

Q12 (a)

3 marks

Question

Evaluate

\int_{3}^{4}(x+2)\sqrt{x-3}\,dx

using the substitution $u=x-3$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Obtains correct anti-derivative in terms of $u$ , or equivalent merit	2
Obtains correct integrand in terms of $u$ , or equivalent merit	1

Sample answer:

\begin{aligned} \int_3^4(x+2)\sqrt{x-3}\,dx&\quad u=x-3,\quad \begin{cases}x=4\to u=1\\x=3\to u=0\end{cases}\\ du&=dx\\ &=\int_0^1(u+5)\sqrt{u}\,du\\ &=\int_0^1\left(u^{3/2}+5u^{1/2}\right)du\\ &=\left[\frac{2}{5}u^{5/2}+\frac{5\times2}{3}u^{3/2}\right]_0^1\\ &=\frac25+\frac{10}{3}\\ &=\frac{56}{15} \end{aligned}

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Syllabus alignment

Calculus

Further Calculus SkillsApply integration by substitution, trigonometric integrals and derivatives of inverse functions, including inverse trigonometric functions.

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