Calculus Examples

$p \int_{1}^{3} {(\frac{1}{x})}^{2} d x$

Step 1

Simplify the expression.

Tap for more steps...

Step 1.1

Apply the product rule to $\frac{1}{x}$ .

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

Step 1.2

One to any power is one.

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

Step 1.3

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

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $- 1$ .

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

Step 2

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

$p - x^{- 1}]_{1}^{3}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Substitute and simplify.

Tap for more steps...

Step 3.1.1

Evaluate $- x^{- 1}$ at $3$ and at $1$ .

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

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

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

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

Step 3.1.2.2

One to any power is one.

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Combine the numerators over the common denominator.

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

Step 3.1.2.5

Add $- 1$ and $3$ .

$p \frac{2}{3}$

Step 3.1.2.6

Combine $p$ and $\frac{2}{3}$ .

$\frac{p \cdot 2}{3}$

Step 3.1.2.7

Move $2$ to the left of $p$ .

$\frac{2 p}{3}$

Step 3.2

Reorder terms.

$\frac{2}{3} p$

Step 4

Combine $\frac{2}{3}$ and $p$ .

$\frac{2 p}{3}$

Step 5