Calculus Examples

$\int \frac{2}{x^{3}} d x$

Step 1

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $- 1$ .

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Simplify.

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

Step 4.3

Simplify.

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Step 4.3.1

Multiply $- 1$ by $2$ .

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

Step 4.3.2

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

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

Step 4.3.3

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

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Step 4.3.3.1

Factor $2$ out of $- 2$ .

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

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.4

Move the negative in front of the fraction.

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