Question

Find a function f that has the given derivative f and value f(c), if possible.

$f\text{'}\left(x\right)=\frac{5}{3x-2},f\left(1\right)=8$

Short answer

The function f is $f\left(x\right)=5\ln \left|3x-2\right|+c$ and the value of (c) is $8$.

Step-by-step solution

Step 1. Given Information.

The derivative:

$$\begin{gathered} f\text{'}\left(x\right)=\frac{5}{3x-2}, \\ f\left(1\right)=8 \end{gathered}$$

Step 2. Find the function by integration.

Integrating both sides with respect to x,

$$\begin{gathered} \int f\text{'}\left(x\right)dx=\int \frac{5}{3x-2}dx \\ =5\int \frac{1}{3x-2}dx \\ =5\ln \left|3x-2\right|+c \end{gathered}$$

where c is a constant.

Step 3. Find the value of c

Substitute the values in the function,

$$\begin{gathered} f\left(x\right)=5\ln \left|3x-2\right|+c \\ 8=5\ln \left|3\left(1\right)-2\right|+c \\ 8=5\ln \left|1\right|+c \\ c=8 \end{gathered}$$