Calculus Examples

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

Step 1

Simplify both sides of the equation.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Combine the numerators over the common denominator.

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

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

Factor the perfect power $1^{2}$ out of $y^{2} + x^{2}$ .

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

Step 1.1.3.2

Factor the perfect power $x^{2}$ out of $x^{2}$ .

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

Step 1.1.3.3

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

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

Step 1.1.4

Pull terms out from under the radical.

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

Step 1.1.5

Combine $\frac{1}{x}$ and $\sqrt{y^{2} + x^{2}}$ .

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

Step 1.2

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

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 2

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

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

Step 3

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

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

Step 4

Reorder $y^{2}$ and $x^{2}$ .

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

Step 5

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

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

Step 6

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

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