Question 3
Determine $\displaystyle2\int(4x+6)^3\,dx$
Not quite. $16=4\times4$, the factor that appears when you differentiate $(4x+6)^4$. Integrating divides by both 4s, but (A) multiplies by them, and the 2 in front has gone missing. The coefficient should be $\dfrac{2}{4\times4}=\dfrac18$.
Not quite. $8=2\times4$: one 4 has been multiplied instead of divided, and the other 4 left out. You get 8 that way whether the 4 you multiply by is the new power or the 4 in $4x$. Integrating divides by both: $\dfrac{2}{4\times4}=\dfrac18$.
Close. $\dfrac12=\dfrac24$ divides by only one 4. You need to divide by the new power, 4, and by 4, the derivative of $4x+6$: $\dfrac{2}{4\times4}=\dfrac18$.
That’s it. Step through the hints below to see why.
Here $16k=2$, so the dashed curve lies right on the black one. Differentiating $\tfrac18(4x+6)^4$ gives back the integrand exactly.
Very close, but isn’t exactly 2, so the curves are close but not equal.
Here , which is times the 2 you need. The dashed curve is too steep, so this $k$ is too big.
Here , less than the 2 you need. The dashed curve is too flat, so this $k$ is too small.
Every curve passes through $(-1.5,\,0)$, where $4x+6=0$, so that point can’t tell them apart.
The 2 is a constant multiple. Leave it in front and integrate $(4x+6)^3$ on its own.
Don’t expand the bracket. The inside, $4x+6$, is linear, so there’s a rule for it.
For a linear inside, raise the power by one, divide by the new power, and divide by the coefficient of $x$.
$\displaystyle\int(ax+b)^n\,dx$$\displaystyle=\frac{(ax+b)^{n+1}}{a(n+1)}+c$
$\displaystyle\int(4x+6)^3\,dx$$\displaystyle=\frac{(4x+6)^4}{4\times4}+c$$\displaystyle=\frac{(4x+6)^4}{16}+c$
Now multiply by the 2.
$2\times\dfrac{(4x+6)^4}{16}$$=\dfrac{(4x+6)^4}{8}$
Check by differentiating: $\dfrac{4\times4}{8}(4x+6)^3=2(4x+6)^3$. The constant is still written $+c$, since twice an arbitrary constant is still arbitrary.
$$\begin{aligned}&2\int(4x+6)^3\,dx\\&=2\times\frac{(4x+6)^4}{4\times4}+c\\&=\frac{(4x+6)^4}{8}+c\end{aligned}$$
Raise the power to 4, divide by the new power and by the 4 in $4x$, then multiply by the 2.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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