Calculus Examples

$\int_{0}^{2} \frac{1}{x^{2}} d x$

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.

$\int_{0}^{2} {(x^{2})}^{- 1} d x$

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

$\int_{0}^{2} x^{2 \cdot - 1} d x$

Step 1.2.2

Multiply $2$ by $- 1$ .

$\int_{0}^{2} x^{- 2} d x$

Step 2

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

$- x^{- 1}]_{0}^{2}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $- x^{- 1}$ at $2$ and at $0$ .

$(- 2^{- 1}) + 0^{- 1}$

Step 3.2

Simplify.

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Step 3.2.1

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

$- \frac{1}{2} + 0^{- 1}$

Step 3.2.2

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

$- \frac{1}{2} + \frac{1}{0}$

Step 3.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{1}{2} + \frac{1}{0}$

Step 3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{1}{2} + \frac{1}{0}$

Step 4

The expression contains a division by $0$ . The expression is undefined.

Undefined