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Numerade Educator

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Problem 13 Easy Difficulty

Find the most general antiderivative of the function.
(Check your answers by differentiation.)

$ f(x) = \dfrac{1}{5} - \dfrac{2}{x} $

Answer

$$\left\{\begin{array}{ll}
\frac{1}{5} x-2 \ln |x|+C_{1} & \text { if } x<0 \\
\frac{1}{5} x-2 \ln |x|+C_{2} & \text { if } x>0
\end{array}\right.$$

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Video Transcript

So for this problem, we want to produce the most general anti derivative of the function And the function that we're given is 1/5 -2 over X. So f of x. because 1 5th -2 over X. And we want to remember that one over X, the anti derivative of that is going to be the natural log of X. So what this is going to give us is 1/5 x minus two times the natural log of the absolute value of acts. All right. There is going to be our final answer, but we want to remember that there is going to be a constant classy. Um and we just know that X cannot equal zero. So it's this right here where X cannot equal zero is supposed to be our final function. If we took a crime of X and chose a different value of C, we see that this is the exact same graph given to us.