Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

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

$- x^{- 1}]_{2}^{5}$

Step 3

Substitute and simplify.

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Step 3.1

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

$(- 5^{- 1}) + 2^{- 1}$

Step 3.2

Simplify.

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Step 3.2.1

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

$- \frac{1}{5} + 2^{- 1}$

Step 3.2.2

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

$- \frac{1}{5} + \frac{1}{2}$

Step 3.2.3

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{5} \cdot \frac{2}{2} + \frac{1}{2}$

Step 3.2.4

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{1}{5} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{5}{5}$

Step 3.2.5

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.5.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$- \frac{2}{5 \cdot 2} + \frac{1}{2} \cdot \frac{5}{5}$

Step 3.2.5.2

Multiply $5$ by $2$ .

$- \frac{2}{10} + \frac{1}{2} \cdot \frac{5}{5}$

Step 3.2.5.3

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$- \frac{2}{10} + \frac{5}{2 \cdot 5}$

Step 3.2.5.4

Multiply $2$ by $5$ .

$- \frac{2}{10} + \frac{5}{10}$

Step 3.2.6

Combine the numerators over the common denominator.

$\frac{- 2 + 5}{10}$

Step 3.2.7

Add $- 2$ and $5$ .

$\frac{3}{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{10}$

Decimal Form:

$0.3$

Step 5