Calculus Examples

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

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.

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

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

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

Substitute and simplify.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

One to any power is one.

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

Step 3.2.3

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

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Add $- 1$ and $5$ .

$\frac{4}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{5}$

Decimal Form:

$0.8$

Step 5