Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $2 y + 2$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1.1

Differentiate $2 y + 2$ .

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

Step 2.2.1.1.2

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

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

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

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

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 $n = 1$ .

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

Step 2.2.1.1.3.3

Multiply $2$ by $1$ .

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

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

Since $2$ is constant with respect to $y$ , the derivative of $2$ 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 $d u$ .

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

Move $2$ to the left of $u$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.2.6

Replace all occurrences of $u$ with $2 y + 2$ .

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

$\ln (| 2 y + 2 |) = 2 (x + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$\ln (| 2 y + 2 |) = 2 x + 2 C$

Step 3.3

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

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

Step 3.4

Rewrite $\ln (| 2 y + 2 |) = 2 x + 2 C$ 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 $b^{y} = x$ .

$e^{2 x + 2 C} = | 2 y + 2 |$

Step 3.5

Solve for $y$ .

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Step 3.5.1

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

$| 2 y + 2 | = e^{2 x + 2 C}$

Step 3.5.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$2 y + 2 = \pm e^{2 x + 2 C}$

Step 3.5.3

Subtract $2$ from both sides of the equation.

$2 y = \pm e^{2 x + 2 C} - 2$

Step 3.5.4

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

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Step 3.5.4.1

Divide each term in $2 y = \pm e^{2 x + 2 C} - 2$ by $2$ .

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $2$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Divide $- 2$ by $2$ .

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

Step 3.5.4.3.2

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

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

Step 3.5.4.3.3

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

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

Step 3.5.4.3.4

Combine the numerators over the common denominator.

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

Step 3.5.4.3.5

Multiply $- 1$ by $2$ .

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

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

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

Step 4.2

Rewrite $e^{2 x + C}$ as $e^{2 x} e^{C}$ .

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

Step 4.3

Reorder $e^{2 x}$ and $e^{C}$ .

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

Step 4.4

Combine constants with the plus or minus.

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