Calculus Examples

$\frac{d y}{d x} (x^{2} + 2) + 4 x (y^{2} + 2 y + 1) = 0$

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.

$\frac{d y}{d x} x^{2} + \frac{d y}{d x} \cdot 2 + 4 x (y^{2} + 2 y + 1) = 0$

Step 1.1.1.2

Move $2$ to the left of $\frac{d y}{d x}$ .

$\frac{d y}{d x} x^{2} + 2 \cdot \frac{d y}{d x} + 4 x (y^{2} + 2 y + 1) = 0$

Step 1.1.1.3

Apply the distributive property.

$\frac{d y}{d x} x^{2} + 2 \frac{d y}{d x} + 4 x y^{2} + 4 x (2 y) + 4 x \cdot 1 = 0$

Step 1.1.1.4

Simplify.

Tap for more steps...

Step 1.1.1.4.1

Rewrite using the commutative property of multiplication.

$\frac{d y}{d x} x^{2} + 2 \frac{d y}{d x} + 4 x y^{2} + 4 \cdot 2 x y + 4 x \cdot 1 = 0$

Step 1.1.1.4.2

Multiply $4$ by $1$ .

$\frac{d y}{d x} x^{2} + 2 \frac{d y}{d x} + 4 x y^{2} + 4 \cdot 2 x y + 4 x = 0$

Step 1.1.1.5

Multiply $4$ by $2$ .

$\frac{d y}{d x} x^{2} + 2 \frac{d y}{d x} + 4 x y^{2} + 8 x y + 4 x = 0$

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

Tap for more steps...

Step 1.1.2.1

Subtract $4 x y^{2}$ from both sides of the equation.

$\frac{d y}{d x} x^{2} + 2 \frac{d y}{d x} + 8 x y + 4 x = - 4 x y^{2}$

Step 1.1.2.2

Subtract $8 x y$ from both sides of the equation.

$\frac{d y}{d x} x^{2} + 2 \frac{d y}{d x} + 4 x = - 4 x y^{2} - 8 x y$

Step 1.1.2.3

Subtract $4 x$ from both sides of the equation.

$\frac{d y}{d x} x^{2} + 2 \frac{d y}{d x} = - 4 x y^{2} - 8 x y - 4 x$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$\frac{d y}{d x} x^{2} + \frac{d y}{d x} \cdot 2 = - 4 x y^{2} - 8 x y - 4 x$

Step 1.1.3.2

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

$\frac{d y}{d x} (x^{2} + 2) = - 4 x y^{2} - 8 x y - 4 x$

Step 1.1.4

Divide each term in $\frac{d y}{d x} (x^{2} + 2) = - 4 x y^{2} - 8 x y - 4 x$ by $x^{2} + 2$ and simplify.

Tap for more steps...

Step 1.1.4.1

Divide each term in $\frac{d y}{d x} (x^{2} + 2) = - 4 x y^{2} - 8 x y - 4 x$ by $x^{2} + 2$ .

$\frac{\frac{d y}{d x} (x^{2} + 2)}{x^{2} + 2} = \frac{- 4 x y^{2}}{x^{2} + 2} + \frac{- 8 x y}{x^{2} + 2} + \frac{- 4 x}{x^{2} + 2}$

Step 1.1.4.2

Simplify the left side.

Tap for more steps...

Step 1.1.4.2.1

Cancel the common factor of $x^{2} + 2$ .

Tap for more steps...

Step 1.1.4.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} (x^{2} + 2)}{x^{2} + 2} = \frac{- 4 x y^{2}}{x^{2} + 2} + \frac{- 8 x y}{x^{2} + 2} + \frac{- 4 x}{x^{2} + 2}$

Step 1.1.4.2.1.2

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

$\frac{d y}{d x} = \frac{- 4 x y^{2}}{x^{2} + 2} + \frac{- 8 x y}{x^{2} + 2} + \frac{- 4 x}{x^{2} + 2}$

Step 1.1.4.3

Simplify the right side.

Tap for more steps...

Step 1.1.4.3.1

Simplify terms.

Tap for more steps...

Step 1.1.4.3.1.1

Simplify each term.

Tap for more steps...

Step 1.1.4.3.1.1.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{4 x y^{2}}{x^{2} + 2} + \frac{- 8 x y}{x^{2} + 2} + \frac{- 4 x}{x^{2} + 2}$

Step 1.1.4.3.1.1.2

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{4 x y^{2}}{x^{2} + 2} - \frac{8 x y}{x^{2} + 2} + \frac{- 4 x}{x^{2} + 2}$

Step 1.1.4.3.1.1.3

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{4 x y^{2}}{x^{2} + 2} - \frac{8 x y}{x^{2} + 2} - \frac{4 x}{x^{2} + 2}$

Step 1.1.4.3.1.2

Combine into one fraction.

Tap for more steps...

Step 1.1.4.3.1.2.1

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{- 4 x y^{2} - 8 x y}{x^{2} + 2} - \frac{4 x}{x^{2} + 2}$

Step 1.1.4.3.1.2.2

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{- 4 x y^{2} - 8 x y - 4 x}{x^{2} + 2}$

Step 1.1.4.3.2

Simplify the numerator.

Tap for more steps...

Step 1.1.4.3.2.1

Factor $4 x$ out of $- 4 x y^{2} - 8 x y - 4 x$ .

Tap for more steps...

Step 1.1.4.3.2.1.1

Factor $4 x$ out of $- 4 x y^{2}$ .

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2}) - 8 x y - 4 x}{x^{2} + 2}$

Step 1.1.4.3.2.1.2

Factor $4 x$ out of $- 8 x y$ .

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2}) + 4 x (- 2 y) - 4 x}{x^{2} + 2}$

Step 1.1.4.3.2.1.3

Factor $4 x$ out of $- 4 x$ .

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2}) + 4 x (- 2 y) + 4 x (- 1)}{x^{2} + 2}$

Step 1.1.4.3.2.1.4

Factor $4 x$ out of $4 x (- 1 y^{2}) + 4 x (- 2 y)$ .

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2} - 2 y) + 4 x (- 1)}{x^{2} + 2}$

Step 1.1.4.3.2.1.5

Factor $4 x$ out of $4 x (- 1 y^{2} - 2 y) + 4 x (- 1)$ .

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2} - 2 y - 1)}{x^{2} + 2}$

Step 1.1.4.3.2.2

Factor by grouping.

Tap for more steps...

Step 1.1.4.3.2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = - 2$ .

Tap for more steps...

Step 1.1.4.3.2.2.1.1

Factor $- 2$ out of $- 2 y$ .

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2} - 2 (y) - 1)}{x^{2} + 2}$

Step 1.1.4.3.2.2.1.2

Rewrite $- 2$ as $- 1$ plus $- 1$

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2} + (- 1 - 1) y - 1)}{x^{2} + 2}$

Step 1.1.4.3.2.2.1.3

Apply the distributive property.

$\frac{d y}{d x} = \frac{4 x (- 1 y^{2} - 1 y - 1 y - 1)}{x^{2} + 2}$

Step 1.1.4.3.2.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.1.4.3.2.2.2.1

Group the first two terms and the last two terms.

$\frac{d y}{d x} = \frac{4 x ((- 1 y^{2} - 1 y) - 1 y - 1)}{x^{2} + 2}$

Step 1.1.4.3.2.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{d y}{d x} = \frac{4 x (y (- 1 y - 1) + 1 (- y - 1))}{x^{2} + 2}$

Step 1.1.4.3.2.2.3

Factor the polynomial by factoring out the greatest common factor, $- 1 y - 1$ .

$\frac{d y}{d x} = \frac{4 x ((- 1 y - 1) (y + 1))}{x^{2} + 2}$

Step 1.1.4.3.2.3

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

$\frac{d y}{d x} = \frac{4 x (- y - 1) (y + 1)}{x^{2} + 2}$

Step 1.1.4.3.2.4

Combine exponents.

Tap for more steps...

Step 1.1.4.3.2.4.1

Factor $- 1$ out of $- y$ .

$\frac{d y}{d x} = \frac{4 x (- (y) - 1) (y + 1)}{x^{2} + 2}$

Step 1.1.4.3.2.4.2

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{d y}{d x} = \frac{4 x (- (y) - 1 (1)) (y + 1)}{x^{2} + 2}$

Step 1.1.4.3.2.4.3

Factor $- 1$ out of $- (y) - 1 (1)$ .

$\frac{d y}{d x} = \frac{4 x (- (y + 1)) (y + 1)}{x^{2} + 2}$

Step 1.1.4.3.2.4.4

Rewrite $- (y + 1)$ as $- 1 (y + 1)$ .

$\frac{d y}{d x} = \frac{4 x (- 1 (y + 1)) (y + 1)}{x^{2} + 2}$

Step 1.1.4.3.2.4.5

Raise $y + 1$ to the power of $1$ .

$\frac{d y}{d x} = \frac{4 x \cdot - 1 ({(y + 1)}^{1} (y + 1))}{x^{2} + 2}$

Step 1.1.4.3.2.4.6

Raise $y + 1$ to the power of $1$ .

$\frac{d y}{d x} = \frac{4 x \cdot - 1 ({(y + 1)}^{1} {(y + 1)}^{1})}{x^{2} + 2}$

Step 1.1.4.3.2.4.7

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

$\frac{d y}{d x} = \frac{4 x \cdot - 1 {(y + 1)}^{1 + 1}}{x^{2} + 2}$

Step 1.1.4.3.2.4.8

Add $1$ and $1$ .

$\frac{d y}{d x} = \frac{4 x \cdot - 1 {(y + 1)}^{2}}{x^{2} + 2}$

Step 1.1.4.3.2.4.9

Multiply $- 1$ by $4$ .

$\frac{d y}{d x} = \frac{- 4 x {(y + 1)}^{2}}{x^{2} + 2}$

Step 1.1.4.3.3

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{4 x {(y + 1)}^{2}}{x^{2} + 2}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = - \frac{4 x}{x^{2} + 2} {(y + 1)}^{2}$

Step 1.3

Multiply both sides by $\frac{1}{{(y + 1)}^{2}}$ .

$\frac{1}{{(y + 1)}^{2}} \frac{d y}{d x} = \frac{1}{{(y + 1)}^{2}} (- \frac{4 x}{x^{2} + 2} {(y + 1)}^{2})$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Rewrite using the commutative property of multiplication.

$\frac{1}{{(y + 1)}^{2}} \frac{d y}{d x} = - \frac{1}{{(y + 1)}^{2}} (\frac{4 x}{x^{2} + 2} {(y + 1)}^{2})$

Step 1.4.2

Cancel the common factor of ${(y + 1)}^{2}$ .

Tap for more steps...

Step 1.4.2.1

Move the leading negative in $- \frac{1}{{(y + 1)}^{2}}$ into the numerator.

$\frac{1}{{(y + 1)}^{2}} \frac{d y}{d x} = \frac{- 1}{{(y + 1)}^{2}} (\frac{4 x}{x^{2} + 2} {(y + 1)}^{2})$

Step 1.4.2.2

Factor ${(y + 1)}^{2}$ out of $\frac{4 x}{x^{2} + 2} {(y + 1)}^{2}$ .

$\frac{1}{{(y + 1)}^{2}} \frac{d y}{d x} = \frac{- 1}{{(y + 1)}^{2}} ({(y + 1)}^{2} \frac{4 x}{x^{2} + 2})$

Step 1.4.2.3

Cancel the common factor.

$\frac{1}{{(y + 1)}^{2}} \frac{d y}{d x} = \frac{- 1}{{(y + 1)}^{2}} ({(y + 1)}^{2} \frac{4 x}{x^{2} + 2})$

Step 1.4.2.4

Rewrite the expression.

$\frac{1}{{(y + 1)}^{2}} \frac{d y}{d x} = - \frac{4 x}{x^{2} + 2}$

Step 1.5

Rewrite the equation.

$\frac{1}{{(y + 1)}^{2}} d y = - \frac{4 x}{x^{2} + 2} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{{(y + 1)}^{2}} d y = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Let $u_{1} = y + 1$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.2.1.1

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

Tap for more steps...

Step 2.2.1.1.1

Differentiate $y + 1$ .

$\frac{d}{d y} [y + 1]$

Step 2.2.1.1.2

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

$\frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.1.1.3

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

$1 + \frac{d}{d y} [1]$

Step 2.2.1.1.4

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

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

$\int \frac{1}{{u_{1}}^{2}} d u_{1} = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.2.1

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

$\int {({u_{1}}^{2})}^{- 1} d u_{1} = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.2.2.2

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

Tap for more steps...

Step 2.2.2.2.1

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

$\int {u_{1}}^{2 \cdot - 1} d u_{1} = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.2.2.2.2

Multiply $2$ by $- 1$ .

$\int {u_{1}}^{- 2} d u_{1} = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.2.3

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

$- {u_{1}}^{- 1} + C_{1} = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.2.4

Rewrite $- {u_{1}}^{- 1} + C_{1}$ as $- \frac{1}{u_{1}} + C_{1}$ .

$- \frac{1}{u_{1}} + C_{1} = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.2.5

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

$- \frac{1}{y + 1} + C_{1} = \int - \frac{4 x}{x^{2} + 2} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

$- \frac{1}{y + 1} + C_{1} = - \int \frac{4 x}{x^{2} + 2} d x$

Step 2.3.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$- \frac{1}{y + 1} + C_{1} = - (4 \int \frac{x}{x^{2} + 2} d x)$

Step 2.3.3

Multiply $4$ by $- 1$ .

$- \frac{1}{y + 1} + C_{1} = - 4 \int \frac{x}{x^{2} + 2} d x$

Step 2.3.4

Let $u_{2} = x^{2} + 2$ . 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 2.3.4.1

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

Tap for more steps...

Step 2.3.4.1.1

Differentiate $x^{2} + 2$ .

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

Step 2.3.4.1.2

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

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

Step 2.3.4.1.3

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

$2 x + \frac{d}{d x} [2]$

Step 2.3.4.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.3.4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.4.2

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

$- \frac{1}{y + 1} + C_{1} = - 4 \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

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

$- \frac{1}{y + 1} + C_{1} = - 4 \int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 2.3.5.2

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

$- \frac{1}{y + 1} + C_{1} = - 4 \int \frac{1}{2 u_{2}} d u_{2}$

Step 2.3.6

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

$- \frac{1}{y + 1} + C_{1} = - 4 (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2})$

Step 2.3.7

Simplify.

Tap for more steps...

Step 2.3.7.1

Combine $\frac{1}{2}$ and $- 4$ .

$- \frac{1}{y + 1} + C_{1} = \frac{- 4}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.7.2

Cancel the common factor of $- 4$ and $2$ .

Tap for more steps...

Step 2.3.7.2.1

Factor $2$ out of $- 4$ .

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

Step 2.3.7.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.7.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.7.2.2.2

Cancel the common factor.

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

Step 2.3.7.2.2.3

Rewrite the expression.

$- \frac{1}{y + 1} + C_{1} = \frac{- 2}{1} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.7.2.2.4

Divide $- 2$ by $1$ .

$- \frac{1}{y + 1} + C_{1} = - 2 \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.8

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

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

Step 2.3.9

Simplify.

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

Step 2.3.10

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y + 1, 1, 1$

Step 3.2.2

Remove parentheses.

$y + 1, 1, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y + 1$

Step 3.3

Multiply each term in $- \frac{1}{y + 1} = - \ln ({| x^{2} + 2 |}^{2}) + C$ by $y + 1$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $y + 1$ .

Tap for more steps...

Step 3.3.2.1.1

Move the leading negative in $- \frac{1}{y + 1}$ into the numerator.

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

Step 3.3.2.1.2

Cancel the common factor.

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

Step 3.3.2.1.3

Rewrite the expression.

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

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.3.1.1

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

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

Step 3.3.3.1.2

Apply the distributive property.

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

Step 3.3.3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.3.3.1.4

Apply the distributive property.

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

Step 3.3.3.1.5

Multiply $C$ by $1$ .

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

Step 3.3.3.2

Reorder factors in $- \ln ({(x^{2} + 2)}^{2}) y - \ln ({(x^{2} + 2)}^{2}) + C y + C$ .

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

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

Rewrite the equation as $- y \ln ({(x^{2} + 2)}^{2}) - \ln ({(x^{2} + 2)}^{2}) + C y + C = - 1$ .

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

Step 3.4.2

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

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

Step 3.4.3

Add $C y$ to both sides of the equation.

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

Step 3.4.4

Add $\ln ({(x^{2} + 2)}^{2})$ to both sides of the equation.

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

Step 3.4.5

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

Tap for more steps...

Step 3.4.5.1

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

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

Step 3.4.5.2

Factor $y$ out of $C y$ .

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

Step 3.4.5.3

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

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

Step 3.4.6

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

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

Step 3.4.7

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

Tap for more steps...

Step 3.4.7.1

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

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

Step 3.4.7.2

Simplify the left side.

Tap for more steps...

Step 3.4.7.2.1

Cancel the common factor of $- \ln ({(x^{2} + 2)}^{2}) + C$ .

Tap for more steps...

Step 3.4.7.2.1.1

Cancel the common factor.

Step 3.4.7.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.7.3

Simplify the right side.

Tap for more steps...

Step 3.4.7.3.1

Simplify each term.

Tap for more steps...

Step 3.4.7.3.1.1

Move the negative in front of the fraction.

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

Step 3.4.7.3.1.2

Move the negative in front of the fraction.

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

Step 3.4.7.3.2

Simplify terms.

Tap for more steps...

Step 3.4.7.3.2.1

Combine the numerators over the common denominator.

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

Step 3.4.7.3.2.2

Combine the numerators over the common denominator.

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

Step 3.4.7.3.2.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 3.4.7.3.2.4

Factor $- 1$ out of $- C - 1 (1)$ .

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

Step 3.4.7.3.2.5

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

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

Step 3.4.7.3.2.6

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

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

Step 3.4.7.3.2.7

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

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

Step 3.4.7.3.2.8

Factor $- 1$ out of $C$ .

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

Step 3.4.7.3.2.9

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

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

Step 3.4.7.3.2.10

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

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

Step 3.4.7.3.2.11

Cancel the common factor.

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

Step 3.4.7.3.2.12

Rewrite the expression.

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

Step 4

Simplify the constant of integration.

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