Determine the following indefinite integral.

$\int \frac{{x}^{5}-3}{x}dx$

Idilwsiw2
2021-11-19
Answered

Determine the following indefinite integral.

$\int \frac{{x}^{5}-3}{x}dx$

You can still ask an expert for help

Theirl1972

Answered 2021-11-20
Author has **22** answers

Step 1

We have to determine the indefinite integral:

$\int \frac{{x}^{5}-3}{x}dx$

Step 2

We can write$\frac{{x}^{5}-3}{x}\text{}as\text{}{x}^{4}-\frac{3}{x}$

So,$\int \frac{{x}^{5}-3}{x}dx=\int ({x}^{4}-\frac{3}{x})dx$

$=\int {x}^{4}dx-\int \frac{3}{x}dx$

$=\frac{{x}^{5}}{5}-3\mathrm{ln}\left(\left|x\right|\right)+C$

Therefore,$\int \frac{{x}^{5}-3}{x}dx=\frac{{x}^{5}}{5}-3\mathrm{ln}\left(\left|x\right|\right)+C$

We have to determine the indefinite integral:

Step 2

We can write

So,

Therefore,

Jennifer Hill

Answered 2021-11-21
Author has **10** answers

Step 1: Simplify $\frac{{x}^{5}-3}{x}\to {x}^{4}-\frac{3}{x}$ .

$\int {x}^{4}-\frac{3}{x}dx$

Step 2: Use Sum Rule:$\int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx$ .

$\int {x}^{4}dx-\int \frac{3}{x}dx$

Step 3: Use Power Rule:$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$ .

$\frac{{x}^{5}}{5}-\int \frac{3}{x}dx$

Step 4: Use Constant Factor Rule:$\int cf\left(x\right)dx=f\int f\left(x\right)dx$ .

$\frac{{x}^{5}}{5}-3\int \frac{1}{x}dx$

Step 5: The derivative of$\mathrm{ln}x\text{}is\text{}\frac{1}{x}$ .

$\frac{{x}^{5}}{5}-3\mathrm{ln}x$

Step 6: Add constant.

$\frac{{x}^{5}}{5}-3\mathrm{ln}x+C$

Step 2: Use Sum Rule:

Step 3: Use Power Rule:

Step 4: Use Constant Factor Rule:

Step 5: The derivative of

Step 6: Add constant.

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