Hint
The first part needs the inner-function divisor 3; the second has new exponent \(\displaystyle \frac{1}{2}\).
Worked solution
(a)
The first term integrates to \(\displaystyle \frac{(3x+ 1)^{3}}{[3\cdot 3]}\). The second gives \(\displaystyle \frac{4x^{\frac{1}{2}}}{\frac{1}{2}}= 8\sqrt{x}\). Thus an antiderivative is \(\displaystyle \frac{(3x+ 1)^{3}}{9}+ 8\sqrt{x}+ C\), for x>0.
Answer: \(\displaystyle \frac{(3x+ 1)^{3}}{9}+ 8\sqrt{x}+ C\).
(b)
The cubic term differentiates to \(\displaystyle (\frac{1}{9})\cdot 3\cdot 3(3x+ 1)^{2}= (3x+ 1)^{2}\). The root term differentiates to 8·(\(\displaystyle \frac{1}{2}\))x⁻¹ᐟ²=4x⁻¹ᐟ² and C differentiates to zero. The given integrand is recovered.
Answer: \(\displaystyle \frac{\frac{d}{dx}\{(3x+ 1)^{3}}{9}+ 8\sqrt{x}+ C\}= (3x+ 1)^{2}+ 4x^{- \frac{1}{2}}\).
(c)
The polynomial part contributes \(\displaystyle \frac{13^{3}- 4^{3}}{9}= \frac{2197- 64}{9}= 237\). The root part contributes \(\displaystyle 8(2-1)=8\). Their total is 245; the integrand is continuous throughout [1,4].
Answer: 245.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Integrate the linear-inner-function square as \(\displaystyle \frac{(3x+ 1)^{3}}{9}\).
- 1 mark: Integrate 4x⁻¹ᐟ² as \(\displaystyle 8\sqrt{x}\).
- 1 mark: Include the arbitrary constant on the stated positive domain.
Alternative methods: Expand the polynomial part first; equivalent constant terms can be absorbed into C.
Common error: Omitting the inner divisor or dividing 4 by the old exponent \(\displaystyle - \frac{1}{2}\).
(b) · 2 marks
- 1 mark: Differentiate the composite cubic with both multipliers to recover \(\displaystyle (3x+1)^{2}\).
- 1 mark: Recover 4x⁻¹ᐟ² and the complete integrand, with the constant differentiating to zero.
Alternative methods: Differentiating an expanded equivalent antiderivative is accepted.
Common error: Checking only one summand or treating the arbitrary constant as a nonzero derivative.
(c) · 3 marks
- 1 mark: Use both endpoints correctly in the composite-cubic part.
- 1 mark: Evaluate the root contribution as 8.
- 1 mark: Give the exact total 245.
Alternative methods: An expanded polynomial antiderivative with exact endpoint evaluation is accepted.
Common error: Using \(\displaystyle x=0\) as a lower limit or substituting into the integrand instead of the antiderivative.
Total: 8 marks.