Calculus Examples

$x (y d) y - (x^{2} + 2) d x = 0$

Step 1

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

$x y d y = (x^{2} + 2) d x$

Step 2

Multiply both sides by $\frac{1}{x}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Split the fraction into multiple fractions.

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

Step 4.3.2

Split the single integral into multiple integrals.

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

Step 4.3.3

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

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Step 4.3.3.1

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

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

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

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

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

Step 4.3.3.2.3

Cancel the common factor.

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

Step 4.3.3.2.4

Rewrite the expression.

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

Step 4.3.3.2.5

Divide $x$ by $1$ .

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

Step 4.3.4

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

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

Step 4.3.5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 4.3.6

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

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

Step 4.3.7

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

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

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

Step 5.2.2.1.2

Apply the distributive property.

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

Step 5.2.2.1.3

Simplify.

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Step 5.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.3.1.1

Cancel the common factor.

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

Step 5.2.2.1.3.1.2

Rewrite the expression.

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

Step 5.2.2.1.3.2

Multiply $2$ by $2$ .

$y^{2} = x^{2} + 4 \ln (| x |) + 2 C$

Step 5.3

Simplify $4 \ln (| x |)$ by moving $4$ inside the logarithm.

$y^{2} = x^{2} + \ln ({| x |}^{4}) + 2 C$

Step 5.4

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

$y = \pm \sqrt{x^{2} + \ln ({| x |}^{4}) + 2 C}$

Step 5.5

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

$y = \pm \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

Step 5.6

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

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Step 5.6.1

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

$y = \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

Step 5.6.3

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

$y = \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

$y = - \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

$y = \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

$y = - \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

$y = \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

$y = - \sqrt{x^{2} + \ln (x^{4}) + 2 C}$

Step 6

Simplify the constant of integration.

$y = \sqrt{x^{2} + \ln (x^{4}) + C}$

$y = - \sqrt{x^{2} + \ln (x^{4}) + C}$