Calculus Examples

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

Step 1

Write $\frac{2 x^{4} + 4 x^{3} - x}{x^{3}}$ as a function.

$f (x) = \frac{2 x^{4} + 4 x^{3} - x}{x^{3}}$

Step 2

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 3

Set up the integral to solve.

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

Step 4

Simplify.

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Step 4.1

Factor $x$ out of $2 x^{4} + 4 x^{3} - x$ .

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Step 4.1.1

Factor $x$ out of $2 x^{4}$ .

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

Step 4.1.2

Factor $x$ out of $4 x^{3}$ .

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

Step 4.1.3

Factor $x$ out of $- x$ .

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

Step 4.1.4

Factor $x$ out of $x (2 x^{3}) + x (4 x^{2})$ .

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

Step 4.1.5

Factor $x$ out of $x (2 x^{3} + 4 x^{2}) + x \cdot - 1$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\int \frac{2 x^{3} + 4 x^{2} - 1}{x^{2}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

Expand $(2 x^{3} + 4 x^{2} - 1) x^{- 2}$ .

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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

Step 6.4

Subtract $2$ from $3$ .

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

Step 6.5

Simplify.

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

Step 6.6

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

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

Step 6.7

Subtract $2$ from $2$ .

$\int 2 x + 4 x^{0} - 1 x^{- 2} d x$

Step 6.8

Anything raised to $0$ is $1$ .

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

Step 6.9

Multiply $4$ by $1$ .

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

Step 6.10

Move $4$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Apply the constant rule.

$2 (\frac{1}{2} x^{2} + C) - (- x^{- 1} + C) + 4 x + C$

Step 13

Simplify.

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Step 13.1

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

$2 (\frac{x^{2}}{2} + C) - (- x^{- 1} + C) + 4 x + C$

Step 13.2

Simplify.

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

Step 14

The answer is the antiderivative of the function $f (x) = \frac{2 x^{4} + 4 x^{3} - x}{x^{3}}$ .

$F (x) =$ $x^{2} + \frac{1}{x} + 4 x + C$