Question 9
Use a suitable double-angle identity to determine $\displaystyle\int 2\sin^2(x)\,dx$.
Not quite. Differentiating this gives $1-4\cos(2x)$. Integrating $\cos(2x)$ divides by 2, it doesn’t multiply by 2. In 2024, 5.6% of Queensland students chose (A).
Not quite. Two things are off: the sign in the identity, and the factor from integrating $\cos(2x)$, which is $\tfrac12$ rather than 2. In 2024, 3.2% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2024, 79.4% of Queensland students chose (C).
Not quite. This would come from $1+\cos(2x)$, which equals $2\cos^2(x)$. For $2\sin^2(x)$ the sign is a minus. In 2024, 11.5% of Queensland students chose (D).
You can’t integrate $\sin^2(x)$ directly. Use the form of the double-angle identity for $\cos(2x)$ that contains only $\sin^2(x)$.
$$\cos(2x)=1-2\sin^2(x)\ \Rightarrow\ 2\sin^2(x)=1-\cos(2x)$$
Now integrate term by term. Integrating $\cos(2x)$ gives $\tfrac12\sin(2x)$, because the chain rule would bring out a 2 when you differentiate back.
$$\int\big(1-\cos(2x)\big)\,dx=x-\frac{\sin(2x)}{2}+c$$
Check any answer by differentiating it. The right option gives back the integrand exactly. On the left, test each option and see which dashed curve sits on the black one.
$$\int 2\sin^2(x)\,dx=\int\big(1-\cos(2x)\big)\,dx=x-\frac{\sin(2x)}{2}+c$$
How Queensland students answered in 2024
From the QCAA Specialist Mathematics subject report for 2024. 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 2024 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2024, 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