Calculus Examples

$\frac{d y}{d x} = \frac{x^{2} y^{3}}{x + 3}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{x^{2}}{x + 3} y^{3}$

Step 1.2

Multiply both sides by $\frac{1}{y^{3}}$ .

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{1}{y^{3}} (\frac{x^{2}}{x + 3} y^{3})$

Step 1.3

Cancel the common factor of $y^{3}$ .

Tap for more steps...

Step 1.3.1

Factor $y^{3}$ out of $\frac{x^{2}}{x + 3} y^{3}$ .

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{1}{y^{3}} (y^{3} \frac{x^{2}}{x + 3})$

Step 1.3.2

Cancel the common factor.

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{1}{y^{3}} (y^{3} \frac{x^{2}}{x + 3})$

Step 1.3.3

Rewrite the expression.

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{x^{2}}{x + 3}$

Step 1.4

Rewrite the equation.

$\frac{1}{y^{3}} d y = \frac{x^{2}}{x + 3} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{3}} d y = \int \frac{x^{2}}{x + 3} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1.1

Move $y^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{3})}^{- 1} d y = \int \frac{x^{2}}{x + 3} d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{3})}^{- 1}$ .

Tap for more steps...

Step 2.2.1.2.1

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

$\int y^{3 \cdot - 1} d y = \int \frac{x^{2}}{x + 3} d x$

Step 2.2.1.2.2

Multiply $3$ by $- 1$ .

$\int y^{- 3} d y = \int \frac{x^{2}}{x + 3} d x$

Step 2.2.2

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

$- \frac{1}{2} y^{- 2} + C_{1} = \int \frac{x^{2}}{x + 3} d x$

Step 2.2.3

Simplify the answer.

Tap for more steps...

Step 2.2.3.1

Rewrite $- \frac{1}{2} y^{- 2} + C_{1}$ as $- \frac{1}{2} \cdot \frac{1}{y^{2}} + C_{1}$ .

$- \frac{1}{2} \cdot \frac{1}{y^{2}} + C_{1} = \int \frac{x^{2}}{x + 3} d x$

Step 2.2.3.2

Simplify.

Tap for more steps...

Step 2.2.3.2.1

Multiply $\frac{1}{y^{2}}$ by $\frac{1}{2}$ .

$- \frac{1}{y^{2} \cdot 2} + C_{1} = \int \frac{x^{2}}{x + 3} d x$

Step 2.2.3.2.2

Move $2$ to the left of $y^{2}$ .

$- \frac{1}{2 y^{2}} + C_{1} = \int \frac{x^{2}}{x + 3} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Divide $x^{2}$ by $x + 3$ .

Tap for more steps...

Step 2.3.1.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{2}$

$0 x$

$0$

Step 2.3.1.2

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

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

Step 2.3.1.3

Multiply the new quotient term by the divisor.

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

Step 2.3.1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 3 x$

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

Step 2.3.1.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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

Step 2.3.1.6

Pull the next terms from the original dividend down into the current dividend.

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

Step 2.3.1.7

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $x$ .

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

Step 2.3.1.8

Multiply the new quotient term by the divisor.

				$x$	-	$3$
$x$	+	$3$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$3 x$
					-	$3 x$	+	$0$
					-	$3 x$	-	$9$

Step 2.3.1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x - 9$

				$x$	-	$3$
$x$	+	$3$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$3 x$
					-	$3 x$	+	$0$
					+	$3 x$	+	$9$

Step 2.3.1.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$	-	$3$
$x$	+	$3$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$3 x$
					-	$3 x$	+	$0$
					+	$3 x$	+	$9$
							+	$9$

Step 2.3.1.11

The final answer is the quotient plus the remainder over the divisor.

$- \frac{1}{2 y^{2}} + C_{1} = \int x - 3 + \frac{9}{x + 3} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$- \frac{1}{2 y^{2}} + C_{1} = \int x d x + \int - 3 d x + \int \frac{9}{x + 3} d x$

Step 2.3.3

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

$- \frac{1}{2 y^{2}} + C_{1} = \frac{1}{2} x^{2} + C_{2} + \int - 3 d x + \int \frac{9}{x + 3} d x$

Step 2.3.4

Apply the constant rule.

$- \frac{1}{2 y^{2}} + C_{1} = \frac{1}{2} x^{2} + C_{2} - 3 x + C_{3} + \int \frac{9}{x + 3} d x$

Step 2.3.5

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$- \frac{1}{2 y^{2}} + C_{1} = \frac{1}{2} x^{2} + C_{2} - 3 x + C_{3} + 9 \int \frac{1}{x + 3} d x$

Step 2.3.6

Let $u = x + 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.3.6.1

Let $u = x + 3$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.3.6.1.1

Differentiate $x + 3$ .

$\frac{d}{d x} [x + 3]$

Step 2.3.6.1.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.6.1.3

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

$1 + \frac{d}{d x} [3]$

Step 2.3.6.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.6.1.5

Add $1$ and $0$ .

$1$

Step 2.3.6.2

Rewrite the problem using $u$ and $d u$ .

$- \frac{1}{2 y^{2}} + C_{1} = \frac{1}{2} x^{2} + C_{2} - 3 x + C_{3} + 9 \int \frac{1}{u} d u$

Step 2.3.7

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

$- \frac{1}{2 y^{2}} + C_{1} = \frac{1}{2} x^{2} + C_{2} - 3 x + C_{3} + 9 (\ln (| u |) + C_{4})$

Step 2.3.8

Simplify.

$- \frac{1}{2 y^{2}} + C_{1} = \frac{1}{2} x^{2} - 3 x + 9 \ln (| u |) + C_{5}$

Step 2.3.9

Replace all occurrences of $u$ with $x + 3$ .

$- \frac{1}{2 y^{2}} + C_{1} = \frac{1}{2} x^{2} - 3 x + 9 \ln (| x + 3 |) + C_{5}$

Step 2.4

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

$- \frac{1}{2 y^{2}} = \frac{1}{2} x^{2} - 3 x + 9 \ln (| x + 3 |) + C$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Simplify $\frac{1}{2} x^{2} - 3 x + 9 \ln (| x + 3 |) + C$ .

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

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

$- \frac{1}{2 y^{2}} = \frac{x^{2}}{2} - 3 x + 9 \ln (| x + 3 |) + C$

Step 3.1.1.2

Simplify $9 \ln (| x + 3 |)$ by moving $9$ inside the logarithm.

$- \frac{1}{2 y^{2}} = \frac{x^{2}}{2} - 3 x + \ln ({| x + 3 |}^{9}) + C$

Step 3.1.2

To write $\ln ({| x + 3 |}^{9})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2}}{2} + \ln ({| x + 3 |}^{9}) \cdot \frac{2}{2} + C$

Step 3.1.3

Simplify terms.

Tap for more steps...

Step 3.1.3.1

Combine $\ln ({| x + 3 |}^{9})$ and $\frac{2}{2}$ .

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2}}{2} + \frac{\ln ({| x + 3 |}^{9}) \cdot 2}{2} + C$

Step 3.1.3.2

Combine the numerators over the common denominator.

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + \ln ({| x + 3 |}^{9}) \cdot 2}{2} + C$

Step 3.1.4

Simplify the numerator.

Tap for more steps...

Step 3.1.4.1

Multiply $\ln ({| x + 3 |}^{9}) \cdot 2$ .

Tap for more steps...

Step 3.1.4.1.1

Reorder $\ln ({| x + 3 |}^{9})$ and $2$ .

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + 2 \cdot \ln ({| x + 3 |}^{9})}{2} + C$

Step 3.1.4.1.2

Simplify $2 \cdot \ln ({| x + 3 |}^{9})$ by moving $2$ inside the logarithm.

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + \ln ({({| x + 3 |}^{9})}^{2})}{2} + C$

Step 3.1.4.2

Multiply the exponents in ${({| x + 3 |}^{9})}^{2}$ .

Tap for more steps...

Step 3.1.4.2.1

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

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + \ln ({| x + 3 |}^{9 \cdot 2})}{2} + C$

Step 3.1.4.2.2

Multiply $9$ by $2$ .

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + \ln ({| x + 3 |}^{18})}{2} + C$

Step 3.1.4.3

Remove the absolute value in ${| x + 3 |}^{18}$ because exponentiations with even powers are always positive.

$- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} + C$

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.

$2 y^{2}, 1, 2, 1$

Step 3.2.2

Since $2 y^{2}, 1, 2, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 1, 2, 1$ then find LCM for the variable part $y^{2}$ .

Step 3.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3.2.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 3.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.2.6

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 3.2.7

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.2.8

The LCM of $2, 1, 2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 3.2.9

The factors for $y^{2}$ are $y \cdot y$ , which is $y$ multiplied by each other $2$ times.

$y^{2} = y \cdot y$

$y$ occurs $2$ times.

Step 3.2.10

The LCM of $y^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y \cdot y$

Step 3.2.11

Multiply $y$ by $y$ .

$y^{2}$

Step 3.2.12

The LCM for $2 y^{2}, 1, 2, 1$ is the numeric part $2$ multiplied by the variable part.

$2 y^{2}$

Step 3.3

Multiply each term in $- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} + C$ by $2 y^{2}$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

Multiply each term in $- \frac{1}{2 y^{2}} = - 3 x + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} + C$ by $2 y^{2}$ .

$- \frac{1}{2 y^{2}} (2 y^{2}) = - 3 x (2 y^{2}) + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} (2 y^{2}) + C (2 y^{2})$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $2 y^{2}$ .

Tap for more steps...

Step 3.3.2.1.1

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

$\frac{- 1}{2 y^{2}} (2 y^{2}) = - 3 x (2 y^{2}) + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} (2 y^{2}) + C (2 y^{2})$

Step 3.3.2.1.2

Cancel the common factor.

$\frac{- 1}{2 y^{2}} (2 y^{2}) = - 3 x (2 y^{2}) + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} (2 y^{2}) + C (2 y^{2})$

Step 3.3.2.1.3

Rewrite the expression.

$- 1 = - 3 x (2 y^{2}) + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} (2 y^{2}) + C (2 y^{2})$

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

Rewrite using the commutative property of multiplication.

$- 1 = - 3 \cdot 2 x y^{2} + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} (2 y^{2}) + C (2 y^{2})$

Step 3.3.3.1.2

Multiply $- 3$ by $2$ .

$- 1 = - 6 x y^{2} + \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} (2 y^{2}) + C (2 y^{2})$

Step 3.3.3.1.3

Rewrite using the commutative property of multiplication.

$- 1 = - 6 x y^{2} + 2 \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} y^{2} + C (2 y^{2})$

Step 3.3.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.3.1.4.1

Cancel the common factor.

$- 1 = - 6 x y^{2} + 2 \frac{x^{2} + \ln ({(x + 3)}^{18})}{2} y^{2} + C (2 y^{2})$

Step 3.3.3.1.4.2

Rewrite the expression.

$- 1 = - 6 x y^{2} + (x^{2} + \ln ({(x + 3)}^{18})) y^{2} + C (2 y^{2})$

Step 3.3.3.1.5

Apply the distributive property.

$- 1 = - 6 x y^{2} + x^{2} y^{2} + \ln ({(x + 3)}^{18}) y^{2} + C (2 y^{2})$

Step 3.3.3.1.6

Rewrite using the commutative property of multiplication.

$- 1 = - 6 x y^{2} + x^{2} y^{2} + \ln ({(x + 3)}^{18}) y^{2} + 2 C y^{2}$

Step 3.3.3.2

Reorder factors in $- 6 x y^{2} + x^{2} y^{2} + \ln ({(x + 3)}^{18}) y^{2} + 2 C y^{2}$ .

$- 1 = - 6 x y^{2} + x^{2} y^{2} + y^{2} \ln ({(x + 3)}^{18}) + 2 C y^{2}$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

Rewrite the equation as $- 6 x y^{2} + x^{2} y^{2} + y^{2} \ln ({(x + 3)}^{18}) + 2 C y^{2} = - 1$ .

$- 6 x y^{2} + x^{2} y^{2} + y^{2} \ln ({(x + 3)}^{18}) + 2 C y^{2} = - 1$

Step 3.4.2

Factor $y^{2}$ out of $- 6 x y^{2} + x^{2} y^{2} + y^{2} \ln ({(x + 3)}^{18}) + 2 C y^{2}$ .

Tap for more steps...

Step 3.4.2.1

Factor $y^{2}$ out of $- 6 x y^{2}$ .

$y^{2} (- 6 x) + x^{2} y^{2} + y^{2} \ln ({(x + 3)}^{18}) + 2 C y^{2} = - 1$

Step 3.4.2.2

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

$y^{2} (- 6 x) + y^{2} x^{2} + y^{2} \ln ({(x + 3)}^{18}) + 2 C y^{2} = - 1$

Step 3.4.2.3

Factor $y^{2}$ out of $y^{2} \ln ({(x + 3)}^{18})$ .

$y^{2} (- 6 x) + y^{2} x^{2} + y^{2} \ln ({(x + 3)}^{18}) + 2 C y^{2} = - 1$

Step 3.4.2.4

Factor $y^{2}$ out of $2 C y^{2}$ .

$y^{2} (- 6 x) + y^{2} x^{2} + y^{2} \ln ({(x + 3)}^{18}) + y^{2} (2 C) = - 1$

Step 3.4.2.5

Factor $y^{2}$ out of $y^{2} (- 6 x) + y^{2} x^{2}$ .

$y^{2} (- 6 x + x^{2}) + y^{2} \ln ({(x + 3)}^{18}) + y^{2} (2 C) = - 1$

Step 3.4.2.6

Factor $y^{2}$ out of $y^{2} (- 6 x + x^{2}) + y^{2} \ln ({(x + 3)}^{18})$ .

$y^{2} (- 6 x + x^{2} + \ln ({(x + 3)}^{18})) + y^{2} (2 C) = - 1$

Step 3.4.2.7

Factor $y^{2}$ out of $y^{2} (- 6 x + x^{2} + \ln ({(x + 3)}^{18})) + y^{2} (2 C)$ .

$y^{2} (- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C) = - 1$

Step 3.4.3

Divide each term in $y^{2} (- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C) = - 1$ by $- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C$ and simplify.

Tap for more steps...

Step 3.4.3.1

Divide each term in $y^{2} (- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C) = - 1$ by $- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C$ .

$\frac{y^{2} (- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C)}{- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C} = \frac{- 1}{- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C}$

Step 3.4.3.2

Simplify the left side.

Tap for more steps...

Step 3.4.3.2.1

Cancel the common factor of $- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C$ .

Tap for more steps...

Step 3.4.3.2.1.1

Cancel the common factor.

$\frac{y^{2} (- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C)}{- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C} = \frac{- 1}{- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C}$

Step 3.4.3.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 1}{- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C}$

Step 3.4.3.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.3.1

Move the negative in front of the fraction.

$y^{2} = - \frac{1}{- 6 x + x^{2} + \ln ({(x + 3)}^{18}) + 2 C}$

Step 3.4.3.3.2

Factor $- 1$ out of $- 6 x$ .

$y^{2} = - \frac{1}{- (6 x) + x^{2} + \ln ({(x + 3)}^{18}) + 2 C}$

Step 3.4.3.3.3

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

$y^{2} = - \frac{1}{- (6 x) - 1 (- x^{2}) + \ln ({(x + 3)}^{18}) + 2 C}$

Step 3.4.3.3.4

Factor $- 1$ out of $- (6 x) - 1 (- x^{2})$ .

$y^{2} = - \frac{1}{- (6 x - x^{2}) + \ln ({(x + 3)}^{18}) + 2 C}$

Step 3.4.3.3.5

Factor $- 1$ out of $\ln ({(x + 3)}^{18})$ .

$y^{2} = - \frac{1}{- (6 x - x^{2}) - 1 (- \ln ({(x + 3)}^{18})) + 2 C}$

Step 3.4.3.3.6

Factor $- 1$ out of $- (6 x - x^{2}) - 1 (- \ln ({(x + 3)}^{18}))$ .

$y^{2} = - \frac{1}{- (6 x - x^{2} - \ln ({(x + 3)}^{18})) + 2 C}$

Step 3.4.3.3.7

Factor $- 1$ out of $2 C$ .

$y^{2} = - \frac{1}{- (6 x - x^{2} - \ln ({(x + 3)}^{18})) - (- 2 C)}$

Step 3.4.3.3.8

Factor $- 1$ out of $- (6 x - x^{2} - \ln ({(x + 3)}^{18})) - (- 2 C)$ .

$y^{2} = - \frac{1}{- (6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}$

Step 3.4.3.3.9

Simplify the expression.

Tap for more steps...

Step 3.4.3.3.9.1

Rewrite $- (6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)$ as $- 1 (6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)$ .

$y^{2} = - \frac{1}{- 1 (6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}$

Step 3.4.3.3.9.2

Move the negative in front of the fraction.

$y^{2} = - - \frac{1}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$

Step 3.4.3.3.9.3

Multiply $- 1$ by $- 1$ .

$y^{2} = 1 \frac{1}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$

Step 3.4.3.3.9.4

Multiply $\frac{1}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$ by $1$ .

$y^{2} = \frac{1}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$

Step 3.4.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{1}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$

Step 3.4.5

Simplify $\pm \sqrt{\frac{1}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$ .

Tap for more steps...

Step 3.4.5.1

Rewrite $\sqrt{\frac{1}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$ as $\frac{\sqrt{1}}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$ .

$y = \pm \frac{\sqrt{1}}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$

Step 3.4.5.2

Any root of $1$ is $1$ .

$y = \pm \frac{1}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$

Step 3.4.5.3

Multiply $\frac{1}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$ by $\frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$ .

$y = \pm \frac{1}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}} \cdot \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$

Step 3.4.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.5.4.1

Multiply $\frac{1}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$ by $\frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$ .

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C} \sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$

Step 3.4.5.4.2

Raise $\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$ to the power of $1$ .

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}^{1} \sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}$

Step 3.4.5.4.3

Raise $\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$ to the power of $1$ .

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}^{1} {\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}^{1}}$

Step 3.4.5.4.4

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

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}^{1 + 1}}$

Step 3.4.5.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}^{2}}$

Step 3.4.5.4.6

Rewrite ${\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}^{2}$ as $6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C$ .

Tap for more steps...

Step 3.4.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$ as ${(6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{({(6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}^{\frac{1}{2}})}^{2}}$

Step 3.4.5.4.6.2

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

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{(6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}^{\frac{1}{2} \cdot 2}}$

Step 3.4.5.4.6.3

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

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{(6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}^{\frac{2}{2}}}$

Step 3.4.5.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.5.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{(6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}^{\frac{2}{2}}}$

Step 3.4.5.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{{(6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C)}^{1}}$

Step 3.4.5.4.6.5

Simplify.

$y = \pm \frac{\sqrt{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}}{6 x - x^{2} - \ln ({(x + 3)}^{18}) - 2 C}$

Step 3.4.6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.4.6.1

First, use the positive value of the $\pm$ to find the first solution.