Calculus Examples

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

Step 1

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

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

Step 2

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 3

Simplify the answer.

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Step 3.1

Simplify.

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Step 3.1.1

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

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

Step 3.1.2

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 3.2

Simplify.

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

Step 3.3

Simplify.

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Step 3.3.1

Multiply $- 1$ by $- 2$ .

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor of $2$ .

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Step 3.3.3.1

Cancel the common factor.

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

Step 3.3.3.2

Rewrite the expression.

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