Calculus Examples

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

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

Step 1

Remove parentheses.

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

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

Step 2

Simplify the expression.

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Step 2.1

Simplify.

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Step 2.1.1

Factor

x

$x$ out of

- x^{2} + x

$- x^{2} + x$ .

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Step 2.1.1.1

Factor

x

$x$ out of

- x^{2}

$- x^{2}$ .

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

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

Step 2.1.1.2

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 2.1.1.3

Factor

x

$x$ out of

x^{1}

$x^{1}$ .

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

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

Step 2.1.1.4

Factor

x

$x$ out of

x (- x) + x \cdot 1

$x (- x) + x \cdot 1$ .

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

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

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

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor

x

$x$ out of

x^{4}

$x^{4}$ .

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

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Move

$x^{3}$ out of the denominator by raising it to the

$- 1$ power.

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

Step 2.2.2

Multiply the exponents in

${(x^{3})}^{- 1}$ .

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Step 2.2.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.2.2

Multiply

$3$ by

$- 1$ .

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

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

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

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

Step 3

Multiply

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

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

Step 4

Simplify.

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Step 4.1

Multiply

$x$ by

$x^{- 3}$ by adding the exponents.

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Step 4.1.1

Move

$x^{- 3}$ .

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

Step 4.1.2

Multiply

$x^{- 3}$ by

$x$ .

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Step 4.1.2.1

Raise

$x$ to the power of

$1$ .

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

Step 4.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4.1.3

Add

$- 3$ and

$1$ .

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

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

Step 4.2

Multiply

$x^{- 3}$ by

$1$ .

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

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

Step 7

By the Power Rule, the integral of

$x^{- 2}$ with respect to

$x$ is

$- x^{- 1}$ .

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

Step 8

By the Power Rule, the integral of

$x^{- 3}$ with respect to

$x$ is

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 9.2.2

Multiply

$\frac{1}{x}$ by

$1$ .

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

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

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

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$