Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Solve for $\frac{d y}{d x}$ .

Tap for more steps...

Step 1.1.1

Simplify each term.

Tap for more steps...

Step 1.1.1.1

Apply the distributive property.

$x^{2} \frac{d y}{d x} - 1 \frac{d y}{d x} + x y = x$

Step 1.1.1.2

Rewrite $- 1 \frac{d y}{d x}$ as $- \frac{d y}{d x}$ .

$x^{2} \frac{d y}{d x} - \frac{d y}{d x} + x y = x$

Step 1.1.2

Subtract $x y$ from both sides of the equation.

$x^{2} \frac{d y}{d x} - \frac{d y}{d x} = x - x y$

Step 1.1.3

Factor $\frac{d y}{d x}$ out of $x^{2} \frac{d y}{d x} - \frac{d y}{d x}$ .

Tap for more steps...

Step 1.1.3.1

Factor $\frac{d y}{d x}$ out of $x^{2} \frac{d y}{d x}$ .

$\frac{d y}{d x} x^{2} - \frac{d y}{d x} = x - x y$

Step 1.1.3.2

Factor $\frac{d y}{d x}$ out of $- \frac{d y}{d x}$ .

$\frac{d y}{d x} x^{2} + \frac{d y}{d x} \cdot - 1 = x - x y$

Step 1.1.3.3

Factor $\frac{d y}{d x}$ out of $\frac{d y}{d x} x^{2} + \frac{d y}{d x} \cdot - 1$ .

$\frac{d y}{d x} (x^{2} - 1) = x - x y$

Step 1.1.4

Rewrite $1$ as $1^{2}$ .

$\frac{d y}{d x} (x^{2} - 1^{2}) = x - x y$

Step 1.1.5

Factor.

Tap for more steps...

Step 1.1.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{d y}{d x} ((x + 1) (x - 1)) = x - x y$

Step 1.1.5.2

Remove unnecessary parentheses.

$\frac{d y}{d x} (x + 1) (x - 1) = x - x y$

Step 1.1.6

Divide each term in $\frac{d y}{d x} (x + 1) (x - 1) = x - x y$ by $(x + 1) (x - 1)$ and simplify.

Tap for more steps...

Step 1.1.6.1

Divide each term in $\frac{d y}{d x} (x + 1) (x - 1) = x - x y$ by $(x + 1) (x - 1)$ .

$\frac{\frac{d y}{d x} (x + 1) (x - 1)}{(x + 1) (x - 1)} = \frac{x}{(x + 1) (x - 1)} + \frac{- x y}{(x + 1) (x - 1)}$

Step 1.1.6.2

Simplify the left side.

Tap for more steps...

Step 1.1.6.2.1

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 1.1.6.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} (x + 1) (x - 1)}{(x + 1) (x - 1)} = \frac{x}{(x + 1) (x - 1)} + \frac{- x y}{(x + 1) (x - 1)}$

Step 1.1.6.2.1.2

Rewrite the expression.

$\frac{\frac{d y}{d x} (x - 1)}{x - 1} = \frac{x}{(x + 1) (x - 1)} + \frac{- x y}{(x + 1) (x - 1)}$

Step 1.1.6.2.2

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 1.1.6.2.2.1

Cancel the common factor.

$\frac{\frac{d y}{d x} (x - 1)}{x - 1} = \frac{x}{(x + 1) (x - 1)} + \frac{- x y}{(x + 1) (x - 1)}$

Step 1.1.6.2.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{x}{(x + 1) (x - 1)} + \frac{- x y}{(x + 1) (x - 1)}$

Step 1.1.6.3

Simplify the right side.

Tap for more steps...

Step 1.1.6.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = \frac{x}{(x + 1) (x - 1)} - \frac{x y}{(x + 1) (x - 1)}$

Step 1.2

Factor.

Tap for more steps...

Step 1.2.1

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{x - x y}{(x + 1) (x - 1)}$

Step 1.2.2

Simplify the numerator.

Tap for more steps...

Step 1.2.2.1

Factor $x$ out of $x - x y$ .

Tap for more steps...

Step 1.2.2.1.1

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

$\frac{d y}{d x} = \frac{x^{1} - x y}{(x + 1) (x - 1)}$

Step 1.2.2.1.2

Factor $x$ out of $x^{1}$ .

$\frac{d y}{d x} = \frac{x \cdot 1 - x y}{(x + 1) (x - 1)}$

Step 1.2.2.1.3

Factor $x$ out of $- x y$ .

$\frac{d y}{d x} = \frac{x \cdot 1 + x (- 1 y)}{(x + 1) (x - 1)}$

Step 1.2.2.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 y)$ .

$\frac{d y}{d x} = \frac{x \cdot (1 - 1 y)}{(x + 1) (x - 1)}$

Step 1.2.2.1.5

Multiply $x$ by $1 - 1 y$ .

$\frac{d y}{d x} = \frac{x (1 - 1 y)}{(x + 1) (x - 1)}$

Step 1.2.2.2

Rewrite $- 1 y$ as $- y$ .

$\frac{d y}{d x} = \frac{x (1 - y)}{(x + 1) (x - 1)}$

Step 1.3

Regroup factors.

$\frac{d y}{d x} = \frac{x}{(x + 1) (x - 1)} (1 - y)$

Step 1.4

Multiply both sides by $\frac{1}{1 - y}$ .

$\frac{1}{1 - y} \frac{d y}{d x} = \frac{1}{1 - y} (\frac{x}{(x + 1) (x - 1)} (1 - y))$

Step 1.5

Cancel the common factor of $1 - y$ .

Tap for more steps...

Step 1.5.1

Factor $1 - y$ out of $\frac{x}{(x + 1) (x - 1)} (1 - y)$ .

$\frac{1}{1 - y} \frac{d y}{d x} = \frac{1}{1 - y} ((1 - y) \frac{x}{(x + 1) (x - 1)})$

Step 1.5.2

Cancel the common factor.

$\frac{1}{1 - y} \frac{d y}{d x} = \frac{1}{1 - y} ((1 - y) \frac{x}{(x + 1) (x - 1)})$

Step 1.5.3

Rewrite the expression.

$\frac{1}{1 - y} \frac{d y}{d x} = \frac{x}{(x + 1) (x - 1)}$

Step 1.6

Rewrite the equation.

$\frac{1}{1 - y} d y = \frac{x}{(x + 1) (x - 1)} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{1 - y} d y = \int \frac{x}{(x + 1) (x - 1)} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Let $u_{1} = 1 - y$ . Then $d u_{1} = - d y$ , so $- d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.2.1.1

Let $u_{1} = 1 - y$ . Find $\frac{d u_{1}}{d y}$ .

Tap for more steps...

Step 2.2.1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{- 1}{u_{1}} d u_{1} = \int \frac{x}{(x + 1) (x - 1)} d x$

Step 2.2.2

Split the fraction into multiple fractions.

$\int - \frac{1}{u_{1}} d u_{1} = \int \frac{x}{(x + 1) (x - 1)} d x$

Step 2.2.3

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

$- \int \frac{1}{u_{1}} d u_{1} = \int \frac{x}{(x + 1) (x - 1)} d x$

Step 2.2.4

The integral of $\frac{1}{u_{1}}$ with respect to $u_{1}$ is $\ln (| u_{1} |)$ .

$- (\ln (| u_{1} |) + C_{1}) = \int \frac{x}{(x + 1) (x - 1)} d x$

Step 2.2.5

Simplify.

$- \ln (| u_{1} |) + C_{1} = \int \frac{x}{(x + 1) (x - 1)} d x$

Step 2.2.6

Replace all occurrences of $u_{1}$ with $1 - y$ .

$- \ln (| 1 - y |) + C_{1} = \int \frac{x}{(x + 1) (x - 1)} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Let $u_{2} = (x + 1) (x - 1)$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 2.3.1.1.1

Differentiate $(x + 1) (x - 1)$ .

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

Step 2.3.1.1.2

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 $f (x) = x + 1$ and $g (x) = x - 1$ .

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

Step 2.3.1.1.3

Differentiate.

Tap for more steps...

Step 2.3.1.1.3.1

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

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

Step 2.3.1.1.3.2

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

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

Step 2.3.1.1.3.3

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

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

Step 2.3.1.1.3.4

Simplify the expression.

Tap for more steps...

Step 2.3.1.1.3.4.1

Add $1$ and $0$ .

$(x + 1) \cdot 1 + (x - 1) \frac{d}{d x} [x + 1]$

Step 2.3.1.1.3.4.2

Multiply $x + 1$ by $1$ .

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

Step 2.3.1.1.3.5

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

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

Step 2.3.1.1.3.6

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

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

Step 2.3.1.1.3.7

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

$x + 1 + (x - 1) (1 + 0)$

Step 2.3.1.1.3.8

Simplify by adding terms.

Tap for more steps...

Step 2.3.1.1.3.8.1

Add $1$ and $0$ .

$x + 1 + (x - 1) \cdot 1$

Step 2.3.1.1.3.8.2

Multiply $x - 1$ by $1$ .

$x + 1 + x - 1$

Step 2.3.1.1.3.8.3

Add $x$ and $x$ .

$2 x + 1 - 1$

Step 2.3.1.1.3.8.4

Simplify by subtracting numbers.

Tap for more steps...

Step 2.3.1.1.3.8.4.1

Subtract $1$ from $1$ .

$2 x + 0$

Step 2.3.1.1.3.8.4.2

Add $2 x$ and $0$ .

$2 x$

Step 2.3.1.2

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

$- \ln (| 1 - y |) + C_{1} = \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{2}$ .

$- \ln (| 1 - y |) + C_{1} = \int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 2.3.2.2

Move $2$ to the left of $u_{2}$ .

$- \ln (| 1 - y |) + C_{1} = \int \frac{1}{2 u_{2}} d u_{2}$

Step 2.3.3

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

$- \ln (| 1 - y |) + C_{1} = \frac{1}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.4

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$- \ln (| 1 - y |) + C_{1} = \frac{1}{2} (\ln (| u_{2} |) + C_{2})$

Step 2.3.5

Simplify.

$- \ln (| 1 - y |) + C_{1} = \frac{1}{2} \ln (| u_{2} |) + C_{2}$

Step 2.3.6

Replace all occurrences of $u_{2}$ with $(x + 1) (x - 1)$ .

$- \ln (| 1 - y |) + C_{1} = \frac{1}{2} \ln (| (x + 1) (x - 1) |) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$- \ln (| 1 - y |) = \frac{1}{2} \ln (| (x + 1) (x - 1) |) + C$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Simplify the right side.

Tap for more steps...

Step 3.1.1

Combine $\frac{1}{2}$ and $\ln (| (x + 1) (x - 1) |)$ .

$- \ln (| 1 - y |) = \frac{\ln (| (x + 1) (x - 1) |)}{2} + C$

Step 3.2

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

$- \ln (| 1 - y |) - \frac{\ln (| (x + 1) (x - 1) |)}{2} = C$

Step 3.3

Simplify the numerator.

Tap for more steps...

Step 3.3.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.1

Apply the distributive property.

$- \ln (| 1 - y |) - \frac{\ln (| x (x - 1) + 1 (x - 1) |)}{2} = C$

Step 3.3.1.2

Apply the distributive property.

$- \ln (| 1 - y |) - \frac{\ln (| x \cdot x + x \cdot - 1 + 1 (x - 1) |)}{2} = C$

Step 3.3.1.3

Apply the distributive property.

$- \ln (| 1 - y |) - \frac{\ln (| x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1 |)}{2} = C$

Step 3.3.2

Simplify and combine like terms.

Tap for more steps...

Step 3.3.2.1

Simplify each term.

Tap for more steps...

Step 3.3.2.1.1

Multiply $x$ by $x$ .

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

Step 3.3.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 3.3.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 3.3.2.1.4

Multiply $x$ by $1$ .

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

Step 3.3.2.1.5

Multiply $- 1$ by $1$ .

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

Step 3.3.2.2

Add $- x$ and $x$ .

$- \ln (| 1 - y |) - \frac{\ln (| x^{2} + 0 - 1 |)}{2} = C$

Step 3.3.2.3

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

$- \ln (| 1 - y |) - \frac{\ln (| x^{2} - 1 |)}{2} = C$

Step 3.4

To write $- \ln (| 1 - y |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \ln (| 1 - y |) \cdot \frac{2}{2} - \frac{\ln (| x^{2} - 1 |)}{2} = C$

Step 3.5

Combine $- \ln (| 1 - y |)$ and $\frac{2}{2}$ .

$\frac{- \ln (| 1 - y |) \cdot 2}{2} - \frac{\ln (| x^{2} - 1 |)}{2} = C$

Step 3.6

Combine the numerators over the common denominator.

$\frac{- \ln (| 1 - y |) \cdot 2 - \ln (| x^{2} - 1 |)}{2} = C$

Step 3.7

Factor $- 1$ out of $- \ln (| 1 - y |) \cdot 2 - \ln (| x^{2} - 1 |)$ .

Tap for more steps...

Step 3.7.1

Reorder the expression.

Tap for more steps...

Step 3.7.1.1

Reorder $1$ and $- y$ .

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

Step 3.7.1.2

Move $\ln (| - y + 1 |)$ .

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

Step 3.7.2

Factor $- 1$ out of $- 1 \cdot 2 \ln (| - y + 1 |)$ .

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

Step 3.7.3

Factor $- 1$ out of $- \ln (| x^{2} - 1 |)$ .

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

Step 3.7.4

Factor $- 1$ out of $- (2 \ln (| - y + 1 |)) - (\ln (| x^{2} - 1 |))$ .

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

Step 3.8

Move the negative in front of the fraction.

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

Step 3.9

Simplify the left side.

Tap for more steps...

Step 3.9.1

Simplify $- \frac{2 \ln (| - y + 1 |) + \ln (| x^{2} - 1 |)}{2}$ .

Tap for more steps...

Step 3.9.1.1

Simplify the numerator.

Tap for more steps...

Step 3.9.1.1.1

Simplify $2 \ln (| - y + 1 |)$ by moving $2$ inside the logarithm.

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

Step 3.9.1.1.2

Remove the absolute value in ${| - y + 1 |}^{2}$ because exponentiations with even powers are always positive.

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

Step 3.9.1.1.3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 3.9.1.2

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

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

Step 3.9.1.3

Simplify $\frac{1}{2} \ln ({(- y + 1)}^{2} | x^{2} - 1 |)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 3.9.1.4

Apply the product rule to ${(- y + 1)}^{2} | x^{2} - 1 |$ .

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

Step 3.9.1.5

Multiply the exponents in ${({(- y + 1)}^{2})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 3.9.1.5.1

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

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

Step 3.9.1.5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.9.1.5.2.1

Cancel the common factor.

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

Step 3.9.1.5.2.2

Rewrite the expression.

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

Step 3.9.1.6

Simplify.

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

Step 3.9.1.7

Apply the distributive property.

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

Step 3.9.1.8

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

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

Step 3.10

Divide each term in $- \ln (- y {| x^{2} - 1 |}^{\frac{1}{2}} + {| x^{2} - 1 |}^{\frac{1}{2}}) = C$ by $- 1$ and simplify.

Tap for more steps...

Step 3.10.1

Divide each term in $- \ln (- y {| x^{2} - 1 |}^{\frac{1}{2}} + {| x^{2} - 1 |}^{\frac{1}{2}}) = C$ by $- 1$ .

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

Step 3.10.2

Simplify the left side.

Tap for more steps...

Step 3.10.2.1

Dividing two negative values results in a positive value.

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

Step 3.10.2.2

Divide $\ln (- y {| x^{2} - 1 |}^{\frac{1}{2}} + {| x^{2} - 1 |}^{\frac{1}{2}})$ by $1$ .

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

Step 3.10.3

Simplify the right side.

Tap for more steps...

Step 3.10.3.1

Move the negative one from the denominator of $\frac{C}{- 1}$ .

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

Step 3.10.3.2

Rewrite $- 1 \cdot C$ as $- C$ .

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

Step 3.11

To solve for $y$ , rewrite the equation using properties of logarithms.

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

Step 3.12

Rewrite $\ln (- y {| x^{2} - 1 |}^{\frac{1}{2}} + {| x^{2} - 1 |}^{\frac{1}{2}}) = - C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- C} = - y {| x^{2} - 1 |}^{\frac{1}{2}} + {| x^{2} - 1 |}^{\frac{1}{2}}$

Step 3.13

Solve for $y$ .

Tap for more steps...

Step 3.13.1

Rewrite the equation as $- y {| x^{2} - 1 |}^{\frac{1}{2}} + {| x^{2} - 1 |}^{\frac{1}{2}} = e^{- C}$ .

$- y {| x^{2} - 1 |}^{\frac{1}{2}} + {| x^{2} - 1 |}^{\frac{1}{2}} = e^{- C}$

Step 3.13.2

Subtract ${| x^{2} - 1 |}^{\frac{1}{2}}$ from both sides of the equation.

$- y {| x^{2} - 1 |}^{\frac{1}{2}} = e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}}$

Step 3.13.3

Divide each term in $- y {| x^{2} - 1 |}^{\frac{1}{2}} = e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}}$ by $- {| x^{2} - 1 |}^{\frac{1}{2}}$ and simplify.

Tap for more steps...

Step 3.13.3.1

Divide each term in $- y {| x^{2} - 1 |}^{\frac{1}{2}} = e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}}$ by $- {| x^{2} - 1 |}^{\frac{1}{2}}$ .

$\frac{- y {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} = \frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.2

Simplify the left side.

Tap for more steps...

Step 3.13.3.2.1

Dividing two negative values results in a positive value.

$\frac{y {| x^{2} - 1 |}^{\frac{1}{2}}}{{| x^{2} - 1 |}^{\frac{1}{2}}} = \frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.2.2

Cancel the common factor.

$\frac{y {| x^{2} - 1 |}^{\frac{1}{2}}}{{| x^{2} - 1 |}^{\frac{1}{2}}} = \frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.2.3

Divide $y$ by $1$ .

$y = \frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3

Simplify the right side.

Tap for more steps...

Step 3.13.3.3.1

To write $\frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$y = \frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} \cdot \frac{- 1}{- 1} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.2

To write $\frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$y = \frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} \cdot \frac{- 1}{- 1} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} \cdot \frac{- 1}{- 1}$

Step 3.13.3.3.3

Write each expression with a common denominator of ${| x^{2} - 1 |}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.13.3.3.3.1

Multiply $\frac{e^{- C}}{- {| x^{2} - 1 |}^{\frac{1}{2}}}$ by $\frac{- 1}{- 1}$ .

$y = \frac{e^{- C} \cdot - 1}{- {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} \cdot \frac{- 1}{- 1}$

Step 3.13.3.3.3.2

Multiply $- 1$ by $- 1$ .

$y = \frac{e^{- C} \cdot - 1}{1 {| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} \cdot \frac{- 1}{- 1}$

Step 3.13.3.3.3.3

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

$y = \frac{e^{- C} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}}}{- {| x^{2} - 1 |}^{\frac{1}{2}}} \cdot \frac{- 1}{- 1}$

Step 3.13.3.3.3.4

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

$y = \frac{e^{- C} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1}{- {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1}$

Step 3.13.3.3.3.5

Multiply $- 1$ by $- 1$ .

$y = \frac{e^{- C} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1}{1 {| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.3.6

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

$y = \frac{e^{- C} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}} + \frac{- {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.4

Combine the numerators over the common denominator.

$y = \frac{e^{- C} \cdot - 1 - {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.5

Simplify each term.

Tap for more steps...

Step 3.13.3.3.5.1

Move $- 1$ to the left of $e^{- C}$ .

$y = \frac{- 1 \cdot e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.5.2

Rewrite $- 1 e^{- C}$ as $- e^{- C}$ .

$y = \frac{- e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.5.3

Multiply $- {| x^{2} - 1 |}^{\frac{1}{2}} \cdot - 1$ .

Tap for more steps...

Step 3.13.3.3.5.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- e^{- C} + 1 {| x^{2} - 1 |}^{\frac{1}{2}}}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.5.3.2

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

$y = \frac{- e^{- C} + {| x^{2} - 1 |}^{\frac{1}{2}}}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.6

Simplify with factoring out.

Tap for more steps...

Step 3.13.3.3.6.1

Factor $- 1$ out of ${| x^{2} - 1 |}^{\frac{1}{2}}$ .

$y = \frac{- e^{- C} - 1 (- {| x^{2} - 1 |}^{\frac{1}{2}})}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.6.2

Factor $- 1$ out of $- e^{- C} - 1 (- {| x^{2} - 1 |}^{\frac{1}{2}})$ .

$y = \frac{- (e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}})}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.6.3

Simplify the expression.

Tap for more steps...

Step 3.13.3.3.6.3.1

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

$y = \frac{- 1 (e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}})}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 3.13.3.3.6.3.2

Move the negative in front of the fraction.

$y = - \frac{e^{- C} - {| x^{2} - 1 |}^{\frac{1}{2}}}{{| x^{2} - 1 |}^{\frac{1}{2}}}$

Step 4

Simplify the constant of integration.

$y = - \frac{C - {| x^{2} - 1 |}^{\frac{1}{2}}}{{| x^{2} - 1 |}^{\frac{1}{2}}}$