Calculus Examples

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

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Subtract $2$ from $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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