Calculus Examples

$\int_{1}^{5} 8 x^{- 2} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int_{1}^{5} x^{- 2} d x$

Step 2

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

$8 (- x^{- 1}]_{1}^{5})$

Step 3

Substitute and simplify.

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Step 3.1

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

$8 ((- 5^{- 1}) + 1^{- 1})$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 (- \frac{1}{5} + 1^{- 1})$

Step 3.2.2

One to any power is one.

$8 (- \frac{1}{5} + 1)$

Step 3.2.3

Write $1$ as a fraction with a common denominator.

$8 (- \frac{1}{5} + \frac{5}{5})$

Step 3.2.4

Combine the numerators over the common denominator.

$8 \frac{- 1 + 5}{5}$

Step 3.2.5

Add $- 1$ and $5$ .

$8 (\frac{4}{5})$

Step 3.2.6

Combine $8$ and $\frac{4}{5}$ .

$\frac{8 \cdot 4}{5}$

Step 3.2.7

Multiply $8$ by $4$ .

$\frac{32}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{5}$

Decimal Form:

$6.4$

Mixed Number Form:

$6 \frac{2}{5}$

Step 5