Calculus Examples

$f (x) = \frac{x^{5} - x^{3} + 6 x}{x^{4}}$

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.

$F (x) = \int \frac{x^{5} - x^{3} + 6 x}{x^{4}} d x$

Step 3

Simplify.

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Step 3.1

Factor $x$ out of $x^{5} - x^{3} + 6 x$ .

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Step 3.1.1

Factor $x$ out of $x^{5}$ .

$\int \frac{x \cdot x^{4} - x^{3} + 6 x}{x^{4}} d x$

Step 3.1.2

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

$\int \frac{x \cdot x^{4} + x (- x^{2}) + 6 x}{x^{4}} d x$

Step 3.1.3

Factor $x$ out of $6 x$ .

$\int \frac{x \cdot x^{4} + x (- x^{2}) + x \cdot 6}{x^{4}} d x$

Step 3.1.4

Factor $x$ out of $x \cdot x^{4} + x (- x^{2})$ .

$\int \frac{x (x^{4} - x^{2}) + x \cdot 6}{x^{4}} d x$

Step 3.1.5

Factor $x$ out of $x (x^{4} - x^{2}) + x \cdot 6$ .

$\int \frac{x (x^{4} - x^{2} + 6)}{x^{4}} d x$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x$ out of $x^{4}$ .

$\int \frac{x (x^{4} - x^{2} + 6)}{x \cdot x^{3}} d x$

Step 3.2.2

Cancel the common factor.

$\int \frac{x (x^{4} - x^{2} + 6)}{x \cdot x^{3}} d x$

Step 3.2.3

Rewrite the expression.

$\int \frac{x^{4} - x^{2} + 6}{x^{3}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int (x^{4} - x^{2} + 6) {(x^{3})}^{- 1} d x$

Step 4.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 4.2.1

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

$\int (x^{4} - x^{2} + 6) x^{3 \cdot - 1} d x$

Step 4.2.2

Multiply $3$ by $- 1$ .

$\int (x^{4} - x^{2} + 6) x^{- 3} d x$

Step 5

Expand $(x^{4} - x^{2} + 6) x^{- 3}$ .

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Step 5.1

Apply the distributive property.

$\int (x^{4} - x^{2}) x^{- 3} + 6 x^{- 3} d x$

Step 5.2

Apply the distributive property.

$\int x^{4} x^{- 3} - x^{2} x^{- 3} + 6 x^{- 3} d x$

Step 5.3

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

$\int x^{4 - 3} - x^{2} x^{- 3} + 6 x^{- 3} d x$

Step 5.4

Subtract $3$ from $4$ .

$\int x^{1} - x^{2} x^{- 3} + 6 x^{- 3} d x$

Step 5.5

Simplify.

$\int x - x^{2} x^{- 3} + 6 x^{- 3} d x$

Step 5.6

Factor out negative.

$\int x - (x^{2} x^{- 3}) + 6 x^{- 3} d x$

Step 5.7

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

$\int x - x^{2 - 3} + 6 x^{- 3} d x$

Step 5.8

Subtract $3$ from $2$ .

$\int x - x^{- 1} + 6 x^{- 3} d x$

Step 6

Split the single integral into multiple integrals.

$\int x d x + \int - x^{- 1} d x + \int 6 x^{- 3} d x$

Step 7

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

$\frac{1}{2} x^{2} + C + \int - x^{- 1} d x + \int 6 x^{- 3} d x$

Step 8

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

$\frac{1}{2} x^{2} + C - \int x^{- 1} d x + \int 6 x^{- 3} d x$

Step 9

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{2} x^{2} + C - (\ln (| x |) + C) + \int 6 x^{- 3} d x$

Step 10

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{2} x^{2} + C - (\ln (| x |) + C) + 6 \int x^{- 3} d x$

Step 11

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$\frac{1}{2} x^{2} + C - (\ln (| x |) + C) + 6 (- \frac{1}{2} x^{- 2} + C)$

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$\frac{1}{2} x^{2} + C - (\ln (| x |) + C) + 6 (- \frac{x^{- 2}}{2} + C)$

Step 12.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} x^{2} + C - (\ln (| x |) + C) + 6 (- \frac{1}{2 x^{2}} + C)$

Step 12.2

Simplify.

$\frac{1}{2} x^{2} - \ln (| x |) + 6 (- \frac{1}{2 x^{2}}) + C$

Step 12.3

Simplify.

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Step 12.3.1

Multiply $- 1$ by $6$ .

$\frac{1}{2} x^{2} - \ln (| x |) - 6 \frac{1}{2 x^{2}} + C$

Step 12.3.2

Combine $- 6$ and $\frac{1}{2 x^{2}}$ .

$\frac{1}{2} x^{2} - \ln (| x |) + \frac{- 6}{2 x^{2}} + C$

Step 12.3.3

Cancel the common factor of $- 6$ and $2$ .

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Step 12.3.3.1

Factor $2$ out of $- 6$ .

$\frac{1}{2} x^{2} - \ln (| x |) + \frac{2 \cdot - 3}{2 x^{2}} + C$

Step 12.3.3.2

Cancel the common factors.

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Step 12.3.3.2.1

Factor $2$ out of $2 x^{2}$ .

$\frac{1}{2} x^{2} - \ln (| x |) + \frac{2 \cdot - 3}{2 (x^{2})} + C$

Step 12.3.3.2.2

Cancel the common factor.

$\frac{1}{2} x^{2} - \ln (| x |) + \frac{2 \cdot - 3}{2 x^{2}} + C$

Step 12.3.3.2.3

Rewrite the expression.

$\frac{1}{2} x^{2} - \ln (| x |) + \frac{- 3}{x^{2}} + C$

Step 12.3.4

Move the negative in front of the fraction.

$\frac{1}{2} x^{2} - \ln (| x |) - \frac{3}{x^{2}} + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{x^{5} - x^{3} + 6 x}{x^{4}}$ .

$F (x) =$ $\frac{1}{2} x^{2} - \ln (| x |) - \frac{3}{x^{2}} + C$