Question 1
Determine $\displaystyle\int 10.4\,x^3\,dx$
That’s it. Step through the hints below to see why. In 2025, 96.7% of Queensland students chose (A).
Not quite. $6.4$ is $10.4-4$. Integrating divides the coefficient by the new power. It doesn’t subtract it: $10.4\div4=2.6$. In 2025, 1.1% of Queensland students chose (B).
Not quite. $14.4$ is $10.4+4$. You add 1 to the power, but the coefficient is divided by the new power: $10.4\div4=2.6$. In 2025, 0.8% of Queensland students chose (C).
Not quite. $41.6$ is $10.4\times4$. Multiplying by the power is what differentiation does. Integration runs the other way, so divide by the new power: $10.4\div4=2.6$. In 2025, 1.3% of Queensland students chose (D).
Differentiating $2.6x^4+c$ gives back exactly $10.4x^3$. The dashed curve sits right on top of the black one, at every $x$.
Differentiating this answer gives , not $10.4x^3$. The dashed curve is times as high as the black one at every $x$, so this answer can’t be the integral.
For a power of $x$, integrate in two moves. First, add one to the power.
$$x^3\ \longrightarrow\ x^{3+1}=x^4$$
The coefficient $10.4$ just comes along for the ride.
Second, divide by the new power. Then add the constant of integration.
$\displaystyle \int 10.4\,x^3\,dx=\frac{10.4\,x^{4}}{4}+c$$\displaystyle =2.6\,x^4+c$
Check by differentiating your answer. You should get the expression you integrated back.
$\displaystyle \frac{d}{dx}\left(2.6\,x^4+c\right)$$\displaystyle =4\times2.6\,x^3$$\displaystyle =10.4\,x^3$
Test each option in the figure. Only one dashed curve lands on the black one.
$\displaystyle \int 10.4\,x^3\,dx=\frac{10.4\,x^{3+1}}{3+1}+c$$\displaystyle =\frac{10.4\,x^4}{4}+c$$\displaystyle =2.6\,x^4+c$
Check: $\dfrac{d}{dx}\left(2.6x^4+c\right)=10.4x^3$.
How Queensland students answered in 2025
From the QCAA Mathematical Methods subject report for 2025. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
Spot a mistake on this page? Report it