Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $x^{- 2}$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

Move $x^{- 2}$ .

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

Step 3.2.2

Multiply $x^{- 2}$ by $x$ .

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Step 3.2.2.1

Raise $x$ to the power of $1$ .

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

Step 3.2.2.2

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

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

Step 3.2.3

Add $- 2$ and $1$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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