Question 3
Determine $\displaystyle\int 10e^{4x}\,dx$
Not quite. This uses the power rule: add 1 to the exponent and divide by the new exponent. That rule is for powers of $x$ like $x^n$. For $e^{4x}$ the exponent stays as it is, and you divide by 4, the coefficient of $x$. In 2021, 8.6% of Queensland students chose (A).
Not quite. This multiplies by 4, which is what differentiating does: $\dfrac{d}{dx}\left(10e^{4x}\right)=40e^{4x}$. Integrating undoes that, so divide by 4 instead. In 2021, 8.0% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2021, 77.0% of Queensland students chose (C).
Not quite. This treats $e^{4x}$ as if it were $x^4$: it raises the 4 to 5 and divides 10 by 5. The exponent of $e$ doesn’t change when you integrate $e^{4x}$. Differentiate $2e^{5x}$ and you get $10e^{5x}$. In 2021, 6.2% of Queensland students chose (D).
Differentiating $\dfrac52e^{4x}+c$ gives back exactly $10e^{4x}$, because $\dfrac52\times4=10$. The dashed curve sits right on top of the black one at every $x$.
(A) isn’t defined at $x=-\tfrac14$, because its denominator $4x+1$ is zero there. $10e^{4x}$ is defined for every $x$, so (A) can’t be its integral.
At this $x$ both give the same value, so checking this one point would fool you. Move the guide. Everywhere else on this graph, the dashed curve misses the black one.
The curves are very close here, but the values in the readout are not equal. Slide the guide to the right end and the gap is clear.
Differentiating (A) needs the quotient rule, and the result, in the readout above, is not $10e^{4x}$. The dashed curve misses the black one here, so (A) isn’t the integral.
Differentiating $40e^{4x}+c$ gives $160e^{4x}$. The dashed curve is 16 times as high as the black one at every $x$, so (B) isn’t the integral.
Differentiating $2e^{5x}+c$ gives $10e^{5x}$, not $10e^{4x}$. The dashed curve misses the black one here, so (D) isn’t the integral.
Differentiating $e^{4x}$ multiplies by 4, the derivative of the inside: $\dfrac{d}{dx}\,e^{4x}=4e^{4x}$. Integrating undoes that, so it divides by 4.
$$\int e^{ax}\,dx=\frac{1}{a}e^{ax}+c$$
The 10 is a constant, so it stays out the front. Then divide by the 4.
$\displaystyle\int10e^{4x}\,dx$$=10\times\dfrac14e^{4x}+c$$=\dfrac52e^{4x}+c$
Check by differentiating. You should get back exactly what you integrated.
$\dfrac{d}{dx}\left(\dfrac52e^{4x}\right)$$=\dfrac52\times4e^{4x}=10e^{4x}$
(B) multiplies by 4 instead of dividing. (A) and (D) treat the exponent as if it were a power of $x$. Test each one in the figure.
$$\begin{aligned}\int10e^{4x}\,dx&=\frac{10}{4}e^{4x}+c\\&=\frac52e^{4x}+c\end{aligned}$$
Divide by 4, the coefficient of $x$ in the exponent. Check: $\dfrac{d}{dx}\left(\dfrac52e^{4x}\right)=10e^{4x}$.
How Queensland students answered in 2021
From the QCAA Mathematical Methods subject report for 2021. 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 2021 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2021, 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