We have an answer to "Determine the following indefinite integral. ∫x5−3xdx"

Idilwsiw2

Answered question

2021-11-19

Determine the following indefinite integral.

\int \frac{x^{5} - 3}{x} d x

Theirl1972

Beginner2021-11-20Added 22 answers

Step 1
We have to determine the indefinite integral:

\int \frac{x^{5} - 3}{x} d x

Step 2
We can write

\frac{x^{5} - 3}{x} a s x^{4} - \frac{3}{x}

So,

\int \frac{x^{5} - 3}{x} d x = \int (x^{4} - \frac{3}{x}) d x

= \int x^{4} d x - \int \frac{3}{x} d x

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

Therefore,

\int \frac{x^{5} - 3}{x} d x = \frac{x^{5}}{5} - 3 \ln (| x |) + C

Jennifer Hill

Beginner2021-11-21Added 10 answers

Step 1: Simplify

\frac{x^{5} - 3}{x} \to x^{4} - \frac{3}{x}

\int x^{4} - \frac{3}{x} d x

Step 2: Use Sum Rule:

\int f (x) + g (x) d x = \int f (x) d x + \int g (x) d x

\int x^{4} d x - \int \frac{3}{x} d x

Step 3: Use Power Rule:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

\frac{x^{5}}{5} - \int \frac{3}{x} d x

Step 4: Use Constant Factor Rule:

\int c f (x) d x = f \int f (x) d x

\frac{x^{5}}{5} - 3 \int \frac{1}{x} d x

Step 5: The derivative of

\ln x i s \frac{1}{x}

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

Step 6: Add constant.

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

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