Calculus Examples

$\int_{- 5}^{5} \frac{2}{x^{3}} d x$

Step 1

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

$2 \int_{- 5}^{5} \frac{1}{x^{3}} d x$

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.

$2 \int_{- 5}^{5} {(x^{3})}^{- 1} d x$

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

$2 \int_{- 5}^{5} x^{3 \cdot - 1} d x$

Step 2.2.2

Multiply $3$ by $- 1$ .

$2 \int_{- 5}^{5} x^{- 3} d x$

Step 3

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

$2 (- \frac{1}{2} x^{- 2}]_{- 5}^{5})$

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 $\frac{1}{2}$ .

$2 (- \frac{x^{- 2}}{2}]_{- 5}^{5})$

Step 4.1.2

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

$2 (- \frac{1}{2 x^{2}}]_{- 5}^{5})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $- \frac{1}{2 x^{2}}$ at $5$ and at $- 5$ .

$2 ((- \frac{1}{2 \cdot 5^{2}}) + \frac{1}{2 {(- 5)}^{2}})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $5$ to the power of $2$ .

$2 (- \frac{1}{2 \cdot 25} + \frac{1}{2 {(- 5)}^{2}})$

Step 4.2.2.2

Multiply $2$ by $25$ .

$2 (- \frac{1}{50} + \frac{1}{2 {(- 5)}^{2}})$

Step 4.2.2.3

Raise $- 5$ to the power of $2$ .

$2 (- \frac{1}{50} + \frac{1}{2 \cdot 25})$

Step 4.2.2.4

Multiply $2$ by $25$ .

$2 (- \frac{1}{50} + \frac{1}{50})$

Step 4.2.2.5

Add $- \frac{1}{50}$ and $\frac{1}{50}$ .

$2 \cdot 0$

Step 4.2.2.6

Multiply $2$ by $0$ .

$0$

Step 5