Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 2 y - 1$ . Then $d u = 2 d y$ , so $\frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 2 y - 1$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $2 y - 1$ .

$\frac{d}{d y} [2 y - 1]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $2 y - 1$ with respect to $y$ is $\frac{d}{d y} [2 y] + \frac{d}{d y} [- 1]$ .

$\frac{d}{d y} [2 y] + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d y} [2 y]$ .

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Step 2.2.1.1.3.1

Since $2$ is constant with respect to $y$ , the derivative of $2 y$ with respect to $y$ is $2 \frac{d}{d y} [y]$ .

$2 \frac{d}{d y} [y] + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3.2

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

$2 \cdot 1 + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d y} [- 1]$

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

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

$2 + 0$

Step 2.2.1.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2.1.2

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

$\int \frac{1}{u^{2}} \cdot \frac{1}{2} d u = \int x^{2} d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

$\int \frac{1}{u^{2} \cdot 2} d u = \int x^{2} d x$

Step 2.2.2.2

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

$\int \frac{1}{2 u^{2}} d u = \int x^{2} d x$

Step 2.2.3

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

$\frac{1}{2} \int \frac{1}{u^{2}} d u = \int x^{2} d x$

Step 2.2.4

Apply basic rules of exponents.

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

$\frac{1}{2} \int u^{2 \cdot - 1} d u = \int x^{2} d x$

Step 2.2.4.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} \int u^{- 2} d u = \int x^{2} d x$

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

Rewrite $\frac{1}{2} (- u^{- 1} + C_{1})$ as $\frac{1}{2} (- \frac{1}{u}) + C_{1}$ .

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

Step 2.2.6.2

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

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

Step 2.2.7

Replace all occurrences of $u$ with $2 y - 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 3.2

Find the LCD of the terms in the equation.

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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.

$2 (2 y - 1), 3, 1$

Step 3.2.2

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 3.2.3

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

$2$ is a prime number

Step 3.2.4

Since $3$ has no factors besides $1$ and $3$ .

$3$ 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

The LCM of $2, 3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 3$

Step 3.2.7

Multiply $2$ by $3$ .

$6$

Step 3.2.8

The factor for $2 y - 1$ is $2 y - 1$ itself.

$(2 y - 1) = 2 y - 1$

$(2 y - 1)$ occurs $1$ time.

Step 3.2.9

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

$2 y - 1$

Step 3.2.10

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$6 (2 y - 1)$

Step 3.3

Multiply each term in $- \frac{1}{2 (2 y - 1)} = \frac{x^{3}}{3} + K$ by $6 (2 y - 1)$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $- \frac{1}{2 (2 y - 1)} = \frac{x^{3}}{3} + K$ by $6 (2 y - 1)$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2 (2 y - 1)$ .

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

Factor $2 (2 y - 1)$ out of $6 (2 y - 1)$ .

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

Step 3.3.2.1.3

Cancel the common factor.

$\frac{- 1}{2 (2 y - 1)} (2 (2 y - 1) \cdot 3) = \frac{x^{3}}{3} (6 (2 y - 1)) + K (6 (2 y - 1))$

Step 3.3.2.1.4

Rewrite the expression.

$- 1 \cdot 3 = \frac{x^{3}}{3} (6 (2 y - 1)) + K (6 (2 y - 1))$

Step 3.3.2.2

Multiply $- 1$ by $3$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.1.2

Cancel the common factor of $3$ .

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Step 3.3.3.1.2.1

Factor $3$ out of $6$ .

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

Step 3.3.3.1.2.2

Cancel the common factor.

$- 3 = 3 \cdot 2 \frac{x^{3}}{3} (2 y - 1) + K (6 (2 y - 1))$

Step 3.3.3.1.2.3

Rewrite the expression.

$- 3 = 2 x^{3} (2 y - 1) + K (6 (2 y - 1))$

Step 3.3.3.1.3

Apply the distributive property.

$- 3 = 2 x^{3} (2 y) + 2 x^{3} \cdot - 1 + K (6 (2 y - 1))$

Step 3.3.3.1.4

Rewrite using the commutative property of multiplication.

$- 3 = 2 \cdot 2 x^{3} y + 2 x^{3} \cdot - 1 + K (6 (2 y - 1))$

Step 3.3.3.1.5

Multiply $- 1$ by $2$ .

$- 3 = 2 \cdot 2 x^{3} y - 2 x^{3} + K (6 (2 y - 1))$

Step 3.3.3.1.6

Multiply $2$ by $2$ .

$- 3 = 4 x^{3} y - 2 x^{3} + K (6 (2 y - 1))$

Step 3.3.3.1.7

Rewrite using the commutative property of multiplication.

$- 3 = 4 x^{3} y - 2 x^{3} + 6 K (2 y - 1)$

Step 3.3.3.1.8

Apply the distributive property.

$- 3 = 4 x^{3} y - 2 x^{3} + 6 K (2 y) + 6 K \cdot - 1$

Step 3.3.3.1.9

Rewrite using the commutative property of multiplication.

$- 3 = 4 x^{3} y - 2 x^{3} + 6 \cdot 2 K y + 6 K \cdot - 1$

Step 3.3.3.1.10

Multiply $- 1$ by $6$ .

$- 3 = 4 x^{3} y - 2 x^{3} + 6 \cdot 2 K y - 6 K$

Step 3.3.3.1.11

Multiply $6$ by $2$ .

$- 3 = 4 x^{3} y - 2 x^{3} + 12 K y - 6 K$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $4 x^{3} y - 2 x^{3} + 12 K y - 6 K = - 3$ .

$4 x^{3} y - 2 x^{3} + 12 K y - 6 K = - 3$

Step 3.4.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.4.2.1

Add $2 x^{3}$ to both sides of the equation.

$4 x^{3} y + 12 K y - 6 K = - 3 + 2 x^{3}$

Step 3.4.2.2

Add $6 K$ to both sides of the equation.

$4 x^{3} y + 12 K y = - 3 + 2 x^{3} + 6 K$

Step 3.4.3

Factor $4 y$ out of $4 x^{3} y + 12 K y$ .

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Step 3.4.3.1

Factor $4 y$ out of $4 x^{3} y$ .

$4 y (x^{3}) + 12 K y = - 3 + 2 x^{3} + 6 K$

Step 3.4.3.2

Factor $4 y$ out of $12 K y$ .

$4 y (x^{3}) + 4 y (3 K) = - 3 + 2 x^{3} + 6 K$

Step 3.4.3.3

Factor $4 y$ out of $4 y (x^{3}) + 4 y (3 K)$ .

$4 y (x^{3} + 3 K) = - 3 + 2 x^{3} + 6 K$

Step 3.4.4

Divide each term in $4 y (x^{3} + 3 K) = - 3 + 2 x^{3} + 6 K$ by $4 (x^{3} + 3 K)$ and simplify.

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Step 3.4.4.1

Divide each term in $4 y (x^{3} + 3 K) = - 3 + 2 x^{3} + 6 K$ by $4 (x^{3} + 3 K)$ .

$\frac{4 y (x^{3} + 3 K)}{4 (x^{3} + 3 K)} = \frac{- 3}{4 (x^{3} + 3 K)} + \frac{2 x^{3}}{4 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $4$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{4 y (x^{3} + 3 K)}{4 (x^{3} + 3 K)} = \frac{- 3}{4 (x^{3} + 3 K)} + \frac{2 x^{3}}{4 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.2.1.2

Rewrite the expression.

$\frac{y (x^{3} + 3 K)}{x^{3} + 3 K} = \frac{- 3}{4 (x^{3} + 3 K)} + \frac{2 x^{3}}{4 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.2.2

Cancel the common factor of $x^{3} + 3 K$ .

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Step 3.4.4.2.2.1

Cancel the common factor.

$\frac{y (x^{3} + 3 K)}{x^{3} + 3 K} = \frac{- 3}{4 (x^{3} + 3 K)} + \frac{2 x^{3}}{4 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 3}{4 (x^{3} + 3 K)} + \frac{2 x^{3}}{4 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Simplify each term.

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Step 3.4.4.3.1.1

Move the negative in front of the fraction.

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{2 x^{3}}{4 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.3.1.2

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

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Step 3.4.4.3.1.2.1

Factor $2$ out of $2 x^{3}$ .

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{2 (x^{3})}{4 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.3.1.2.2

Cancel the common factors.

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Step 3.4.4.3.1.2.2.1

Factor $2$ out of $4 (x^{3} + 3 K)$ .

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{2 (x^{3})}{2 (2 (x^{3} + 3 K))} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.3.1.2.2.2

Cancel the common factor.

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{2 x^{3}}{2 (2 (x^{3} + 3 K))} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.3.1.2.2.3

Rewrite the expression.

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{x^{3}}{2 (x^{3} + 3 K)} + \frac{6 K}{4 (x^{3} + 3 K)}$

Step 3.4.4.3.1.3

Cancel the common factor of $6$ and $4$ .

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Step 3.4.4.3.1.3.1

Factor $2$ out of $6 K$ .

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{x^{3}}{2 (x^{3} + 3 K)} + \frac{2 (3 K)}{4 (x^{3} + 3 K)}$

Step 3.4.4.3.1.3.2

Cancel the common factors.

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Step 3.4.4.3.1.3.2.1

Factor $2$ out of $4 (x^{3} + 3 K)$ .

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{x^{3}}{2 (x^{3} + 3 K)} + \frac{2 (3 K)}{2 (2 (x^{3} + 3 K))}$

Step 3.4.4.3.1.3.2.2

Cancel the common factor.

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{x^{3}}{2 (x^{3} + 3 K)} + \frac{2 (3 K)}{2 (2 (x^{3} + 3 K))}$

Step 3.4.4.3.1.3.2.3

Rewrite the expression.

$y = - \frac{3}{4 (x^{3} + 3 K)} + \frac{x^{3}}{2 (x^{3} + 3 K)} + \frac{3 K}{2 (x^{3} + 3 K)}$

Step 4

Simplify the constant of integration.

$y = - \frac{3}{4 (x^{3} + K)} + \frac{x^{3}}{2 (x^{3} + K)} + \frac{K}{2 (x^{3} + K)}$