Gage Potter

2022-05-03

Evaluating $\int (\frac{3}{5}-\frac{8}{x})\text{}dx$

I am unsure how to get the antiderivative of $f(x)=\frac{3}{5}-\frac{8}{x}$

I know the answer is $-8\mathrm{ln}(|x|)+C+\frac{3x}{5}$ as it says in my textbook.

But I am unsure how I get something like ln(x) in the answer? Could someone guide me through this? I am totally lost as for other antiderivatives I have no problem but this stuns me.

I am unsure how to get the antiderivative of $f(x)=\frac{3}{5}-\frac{8}{x}$

I know the answer is $-8\mathrm{ln}(|x|)+C+\frac{3x}{5}$ as it says in my textbook.

But I am unsure how I get something like ln(x) in the answer? Could someone guide me through this? I am totally lost as for other antiderivatives I have no problem but this stuns me.

Brianna Sims

Beginner2022-05-04Added 19 answers

Explanation:

The antiderivative of $\frac{1}{x}$ is $\mathrm{ln}|x|$. You can break up the antiderivatives by addition, subtraction, or you can pull scalar multiples out to the front. Since $-{\displaystyle \frac{8}{x}}$ is just $-8\cdot {\displaystyle \frac{1}{x}}$, the antiderivative is $-8\mathrm{ln}|x|+C$.

The antiderivative of $\frac{1}{x}$ is $\mathrm{ln}|x|$. You can break up the antiderivatives by addition, subtraction, or you can pull scalar multiples out to the front. Since $-{\displaystyle \frac{8}{x}}$ is just $-8\cdot {\displaystyle \frac{1}{x}}$, the antiderivative is $-8\mathrm{ln}|x|+C$.

Lughettitbn

Beginner2022-05-05Added 12 answers

Step 1

You need to use the fact the derivative is linear

${(\textcolor[rgb]{}{k}\cdot {f(x)}+\textcolor[rgb]{}{j}\cdot {g(x)})}^{\prime}=\textcolor[rgb]{}{k}\cdot {{f}^{\prime}(x)}+\textcolor[rgb]{}{j}\cdot {{g}^{\prime}(x)}$

You seem to know that $(x{)}^{\prime}=1$

$(\mathrm{log}x{)}^{\prime}={\displaystyle \frac{1}{x}}$

Step 2

Then, using the above, can you find a primitive of $f(x)=\frac{3}{5}\cdot {1}+(-8)\cdot {\frac{1}{x}}\text{?}$

You need to use the fact the derivative is linear

${(\textcolor[rgb]{}{k}\cdot {f(x)}+\textcolor[rgb]{}{j}\cdot {g(x)})}^{\prime}=\textcolor[rgb]{}{k}\cdot {{f}^{\prime}(x)}+\textcolor[rgb]{}{j}\cdot {{g}^{\prime}(x)}$

You seem to know that $(x{)}^{\prime}=1$

$(\mathrm{log}x{)}^{\prime}={\displaystyle \frac{1}{x}}$

Step 2

Then, using the above, can you find a primitive of $f(x)=\frac{3}{5}\cdot {1}+(-8)\cdot {\frac{1}{x}}\text{?}$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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