Calculus Examples

$\int \frac{1}{x^{3}} \sqrt{1 - \frac{1}{x^{2}}} d x$

Step 1

Combine $\frac{1}{x^{3}}$ and $\sqrt{1 - \frac{1}{x^{2}}}$ .

$\int \frac{\sqrt{1 - \frac{1}{x^{2}}}}{x^{3}} d x$

Step 2

Let $u_{2} = 1 - \frac{1}{x^{2}}$ . Then $d u_{2} = \frac{2}{x^{3}} d x$ , so $- d u_{2} = \frac{- 2}{x^{3}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 2.1

Let $u_{2} = 1 - \frac{1}{x^{2}}$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $1 - \frac{1}{x^{2}}$ .

$\frac{d}{d x} [1 - \frac{1}{x^{2}}]$

Step 2.1.2

Differentiate.

Tap for more steps...

Step 2.1.2.1

By the Sum Rule, the derivative of $1 - \frac{1}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{1}{x^{2}}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 2.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- \frac{1}{x^{2}}]$ .

Tap for more steps...

Step 2.1.3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

$0 - \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.1.3.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$0 - \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

Tap for more steps...

Step 2.1.3.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$0 - (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.1.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = - 1$ .

$0 - (- {u_{1}}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.1.3.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$0 - (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 2.1.3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$0 - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.6

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

Tap for more steps...

Step 2.1.3.6.1

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

$0 - (- x^{2 \cdot - 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.6.2

Multiply $2$ by $- 2$ .

$0 - (- x^{- 4} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.7

Multiply $2$ by $- 1$ .

$0 - (- 2 x^{- 4} x) + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.8

Raise $x$ to the power of $1$ .

$0 - (- 2 (x^{1} x^{- 4})) + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 - (- 2 x^{1 - 4}) + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.10

Subtract $4$ from $1$ .

$0 - (- 2 x^{- 3}) + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.11

Multiply $- 2$ by $- 1$ .

$0 + 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0$

Step 2.1.3.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

$0 + 2 x^{- 3} + 0$

Step 2.1.3.13

Add $2 x^{- 3}$ and $0$ .

$0 + 2 x^{- 3}$

Step 2.1.4

Simplify.

Tap for more steps...

Step 2.1.4.1

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

$0 + 2 \frac{1}{x^{3}}$

Step 2.1.4.2

Combine terms.

Tap for more steps...

Step 2.1.4.2.1

Combine $2$ and $\frac{1}{x^{3}}$ .

$0 + \frac{2}{x^{3}}$

Step 2.1.4.2.2

Add $0$ and $\frac{2}{x^{3}}$ .

$\frac{2}{x^{3}}$

Step 2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{2} \sqrt{u_{2}} d u_{2}$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \sqrt{u_{2}} d u_{2}$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{2}}$ as ${u_{2}}^{\frac{1}{2}}$ .

$\frac{1}{2} \int {u_{2}}^{\frac{1}{2}} d u_{2}$

Step 5

By the Power Rule, the integral of ${u_{2}}^{\frac{1}{2}}$ with respect to $u_{2}$ is $\frac{2}{3} {u_{2}}^{\frac{3}{2}}$ .

$\frac{1}{2} (\frac{2}{3} {u_{2}}^{\frac{3}{2}} + C)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Rewrite $\frac{1}{2} (\frac{2}{3} {u_{2}}^{\frac{3}{2}} + C)$ as $\frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} + C$ .

$\frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} + C$

Step 6.2

Rewrite $\frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} + C$ as $\frac{1}{3} {u_{2}}^{\frac{3}{2}} + C$ .

$\frac{1}{3} {u_{2}}^{\frac{3}{2}} + C$

Step 7

Replace all occurrences of $u_{2}$ with $1 - \frac{1}{x^{2}}$ .

$\frac{1}{3} {(1 - \frac{1}{x^{2}})}^{\frac{3}{2}} + C$