Question

In Exercises 59–63, find a function f that has the given derivative f'. In each case you can find the answer with an educated guess-and-check process.

$f\text{'}\left(x\right)=\frac{x}{x^2+3}$

Short answer

The required function is $f\left(x\right)=\frac{1}{2}\log (x^2+3)$

Step-by-step solution

Step 1. Given Information  

The given derivative is$f\text{'}\left(x\right)=\frac{x}{x^2+3}$

Step 2. Calculation  

We will use a targeted guess and check method to find an antiderivative of given derivative.

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}\log (x^2+3)·\frac{1}{2} \\ =\frac{x}{x^2+3} \end{gathered}$$

Thus, we get the required function.