Calculus Examples

- ln ⎛ ⎝ 1 - \sqrt{\frac{y^{2}}{x^{2}} + 1} ⎞ ⎠ = ln (x) + c

$- \ln (1 - \sqrt{\frac{y^{2}}{x^{2}} + 1}) = \ln (x) + c$

Step 1

Simplify both sides of the equation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Write

1

$1$ as a fraction with a common denominator.

- ln ⎛ ⎝ 1 - \sqrt{\frac{y^{2}}{x^{2}} + \frac{x^{2}}{x^{2}}} ⎞ ⎠ = ln (x) + c

$- \ln (1 - \sqrt{\frac{y^{2}}{x^{2}} + \frac{x^{2}}{x^{2}}}) = \ln (x) + c$

Step 1.1.2

Combine the numerators over the common denominator.

- ln ⎛ ⎝ 1 - \sqrt{\frac{y^{2} + x^{2}}{x^{2}}} ⎞ ⎠ = ln (x) + c

$- \ln (1 - \sqrt{\frac{y^{2} + x^{2}}{x^{2}}}) = \ln (x) + c$

Step 1.1.3

Rewrite

\frac{y^{2} + x^{2}}{x^{2}}

$\frac{y^{2} + x^{2}}{x^{2}}$ as

{(\frac{1}{x})}^{2} (y^{2} + x^{2})

${(\frac{1}{x})}^{2} (y^{2} + x^{2})$ .

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Step 1.1.3.1

Factor the perfect power

1^{2}

$1^{2}$ out of

y^{2} + x^{2}

$y^{2} + x^{2}$ .

- ln ⎛ ⎝ 1 - \sqrt{\frac{1^{2} (y^{2} + x^{2})}{x^{2}}} ⎞ ⎠ = ln (x) + c

$- \ln (1 - \sqrt{\frac{1^{2} (y^{2} + x^{2})}{x^{2}}}) = \ln (x) + c$

Step 1.1.3.2

Factor the perfect power

x^{2}

$x^{2}$ out of

x^{2}

$x^{2}$ .

- ln ⎛ ⎝ 1 - \sqrt{\frac{1^{2} (y^{2} + x^{2})}{x^{2} \cdot 1}} ⎞ ⎠ = ln (x) + c

$- \ln (1 - \sqrt{\frac{1^{2} (y^{2} + x^{2})}{x^{2} \cdot 1}}) = \ln (x) + c$

Step 1.1.3.3

Rearrange the fraction

\frac{1^{2} (y^{2} + x^{2})}{x^{2} \cdot 1}

$\frac{1^{2} (y^{2} + x^{2})}{x^{2} \cdot 1}$ .

- ln ⎛ ⎝ 1 - \sqrt{{(\frac{1}{x})}^{2} (y^{2} + x^{2})} ⎞ ⎠ = ln (x) + c

$- \ln (1 - \sqrt{{(\frac{1}{x})}^{2} (y^{2} + x^{2})}) = \ln (x) + c$

- ln ⎛ ⎝ 1 - \sqrt{{(\frac{1}{x})}^{2} (y^{2} + x^{2})} ⎞ ⎠ = ln (x) + c

$- \ln (1 - \sqrt{{(\frac{1}{x})}^{2} (y^{2} + x^{2})}) = \ln (x) + c$

Step 1.1.4

Pull terms out from under the radical.

- ln (1 - (\frac{1}{x} \sqrt{y^{2} + x^{2}})) = ln (x) + c

$- \ln (1 - (\frac{1}{x} \sqrt{y^{2} + x^{2}})) = \ln (x) + c$

Step 1.1.5

Combine

\frac{1}{x}

$\frac{1}{x}$ and

\sqrt{y^{2} + x^{2}}

$\sqrt{y^{2} + x^{2}}$ .

- ln (1 - \frac{\sqrt{y^{2} + x^{2}}}{x}) = ln (x) + c

$- \ln (1 - \frac{\sqrt{y^{2} + x^{2}}}{x}) = \ln (x) + c$

- ln (1 - \frac{\sqrt{y^{2} + x^{2}}}{x}) = ln (x) + c

$- \ln (1 - \frac{\sqrt{y^{2} + x^{2}}}{x}) = \ln (x) + c$

Step 1.2

Write

1

$1$ as a fraction with a common denominator.

- ln (\frac{x}{x} - \frac{\sqrt{y^{2} + x^{2}}}{x}) = ln (x) + c

$- \ln (\frac{x}{x} - \frac{\sqrt{y^{2} + x^{2}}}{x}) = \ln (x) + c$

Step 1.3

Combine the numerators over the common denominator.

- ln (\frac{x - \sqrt{y^{2} + x^{2}}}{x}) = ln (x) + c

$- \ln (\frac{x - \sqrt{y^{2} + x^{2}}}{x}) = \ln (x) + c$

- ln (\frac{x - \sqrt{y^{2} + x^{2}}}{x}) = ln (x) + c

$- \ln (\frac{x - \sqrt{y^{2} + x^{2}}}{x}) = \ln (x) + c$

Step 2

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{y^{2} + x^{2}}

$\sqrt{y^{2} + x^{2}}$ as

{(y^{2} + x^{2})}^{\frac{1}{2}}

${(y^{2} + x^{2})}^{\frac{1}{2}}$ .

- ln ⎛ ⎜ ⎝ \frac{x - {(y^{2} + x^{2})}^{\frac{1}{2}}}{x} ⎞ ⎟ ⎠ = ln (x) + c

$- \ln (\frac{x - {(y^{2} + x^{2})}^{\frac{1}{2}}}{x}) = \ln (x) + c$

Step 3

Move all the terms containing a logarithm to the left side of the equation.

- ln ⎛ ⎜ ⎝ \frac{x - {(y^{2} + x^{2})}^{\frac{1}{2}}}{x} ⎞ ⎟ ⎠ - ln (x) = c

$- \ln (\frac{x - {(y^{2} + x^{2})}^{\frac{1}{2}}}{x}) - \ln (x) = c$

Step 4

Reorder

y^{2}

$y^{2}$ and

x^{2}

$x^{2}$ .

- ln ⎛ ⎜ ⎝ \frac{x - {(x^{2} + y^{2})}^{\frac{1}{2}}}{x} ⎞ ⎟ ⎠ - ln (x) = c

$- \ln (\frac{x - {(x^{2} + y^{2})}^{\frac{1}{2}}}{x}) - \ln (x) = c$

Step 5

Add

ln (x)

$\ln (x)$ to both sides of the equation.

- ln ⎛ ⎜ ⎝ \frac{x - {(x^{2} + y^{2})}^{\frac{1}{2}}}{x} ⎞ ⎟ ⎠ = c + ln (x)

$- \ln (\frac{x - {(x^{2} + y^{2})}^{\frac{1}{2}}}{x}) = c + \ln (x)$

Step 6

Move all the terms containing a logarithm to the left side of the equation.

- ln ⎛ ⎜ ⎝ \frac{x - {(x^{2} + y^{2})}^{\frac{1}{2}}}{x} ⎞ ⎟ ⎠ - ln (x) = c

$- \ln (\frac{x - {(x^{2} + y^{2})}^{\frac{1}{2}}}{x}) - \ln (x) = c$