Calculus Examples

$\int_{1}^{8} \frac{3}{x^{2}} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{1}^{8} \frac{1}{x^{2}} d x$

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.

$3 \int_{1}^{8} {(x^{2})}^{- 1} d x$

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

$3 \int_{1}^{8} x^{2 \cdot - 1} d x$

Step 2.2.2

Multiply $2$ by $- 1$ .

$3 \int_{1}^{8} x^{- 2} d x$

Step 3

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

$3 (- x^{- 1}]_{1}^{8})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $- x^{- 1}$ at $8$ and at $1$ .

$3 ((- 8^{- 1}) + 1^{- 1})$

Step 4.2

Simplify.

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Step 4.2.1

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

$3 (- \frac{1}{8} + 1^{- 1})$

Step 4.2.2

One to any power is one.

$3 (- \frac{1}{8} + 1)$

Step 4.2.3

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

$3 (- \frac{1}{8} + \frac{8}{8})$

Step 4.2.4

Combine the numerators over the common denominator.

$3 \frac{- 1 + 8}{8}$

Step 4.2.5

Add $- 1$ and $8$ .

$3 (\frac{7}{8})$

Step 4.2.6

Combine $3$ and $\frac{7}{8}$ .

$\frac{3 \cdot 7}{8}$

Step 4.2.7

Multiply $3$ by $7$ .

$\frac{21}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{21}{8}$

Decimal Form:

$2.625$

Mixed Number Form:

$2 \frac{5}{8}$

Step 6