Calculus Examples

$f (x) = \frac{x^{6} - x}{x^{3}}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int \frac{x^{6} - x}{x^{3}} d x$

Step 3

Simplify the expression.

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Step 3.1

Simplify.

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Step 3.1.1

Factor $x$ out of $x^{6} - x$ .

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Step 3.1.1.1

Factor $x$ out of $x^{6}$ .

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

Step 3.1.1.2

Factor $x$ out of $- x$ .

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

Step 3.1.1.3

Factor $x$ out of $x \cdot x^{5} + x \cdot - 1$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $x$ out of $x^{3}$ .

$\int \frac{x (x^{5} - 1)}{x \cdot x^{2}} d x$

Step 3.1.2.2

Cancel the common factor.

$\int \frac{x (x^{5} - 1)}{x \cdot x^{2}} d x$

Step 3.1.2.3

Rewrite the expression.

$\int \frac{x^{5} - 1}{x^{2}} d x$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int (x^{5} - 1) {(x^{2})}^{- 1} d x$

Step 3.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 3.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int (x^{5} - 1) x^{2 \cdot - 1} d x$

Step 3.2.2.2

Multiply $2$ by $- 1$ .

$\int (x^{5} - 1) x^{- 2} d x$

Step 4

Multiply $(x^{5} - 1) x^{- 2}$ .

$\int x^{5} x^{- 2} - 1 x^{- 2} d x$

Step 5

Simplify.

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Step 5.1

Multiply $x^{5}$ by $x^{- 2}$ by adding the exponents.

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Step 5.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{5 - 2} - 1 x^{- 2} d x$

Step 5.1.2

Subtract $2$ from $5$ .

$\int x^{3} - 1 x^{- 2} d x$

Step 5.2

Rewrite $- 1 x^{- 2}$ as $- x^{- 2}$ .

$\int x^{3} - x^{- 2} d x$

Step 6

Split the single integral into multiple integrals.

$\int x^{3} d x + \int - x^{- 2} d x$

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{4} x^{4} + C + \int - x^{- 2} d x$

Step 8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{4} x^{4} + C - \int x^{- 2} d x$

Step 9

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$\frac{1}{4} x^{4} + C - (- x^{- 1} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{4} x^{4} - - x^{- 1} + C$

Step 10.2

Simplify.

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Step 10.2.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{4} x^{4} + 1 x^{- 1} + C$

Step 10.2.2

Multiply $x^{- 1}$ by $1$ .

$\frac{1}{4} x^{4} + x^{- 1} + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{x^{6} - x}{x^{3}}$ .

$F (x) =$ $\frac{1}{4} x^{4} + x^{- 1} + C$