Science:Math Exam Resources/Courses/MATH100/December 2016/Question 06 (a)

MATH100 December 2016

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Question 06 (a)
Which of the following is the most general antiderivative of the function $e^{2x+3}$? In the functions below, $c$ is an arbitrary constant. (i) ${\frac {1}{2}}e^{2x+3}+c$ (ii) $2e^{2x+3}+c$ (iii) $3e^{2x+3}+c$ (iv) ${\frac {1}{3}}e^{2x+3}+c$ (v) $e^{2x+3}+c$ (vi) $(2x+3)e^{2x+2}+c$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The derivative of an antiderivative of a function should, by definition, be the original function.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Note that $(e^{2x+3})'=2e^{2x+3}$. In other words, $\left({\frac {1}{2}}e^{2x+3}\right)'=e^{2x+3}$. Recall that the anti-derivative $F$ of $f$ is a function satisfying $F'=f$. Also, for any constant $c$, we have $(c)'=0$. As a result, the most general anti-derivative of $f(x)=e^{2x+3}$ is $\color {blue}{\frac {1}{2}}e^{2x+3}+c.$ Remarks: one could also complete the question by taking the derivative of all the available anti-derivatives and checking if the result is equal to the original function (which for this question, may seem convenient because the answer is the first given anti-derivative), but this method may take more time, depending on the complexity and order of the given anti-derivatives.