Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $y^{2}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

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

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Step 1.3.2.1

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

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

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

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Step 2.3.3.1

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

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

Step 2.3.3.2

Cancel the common factors.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

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

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

Step 2.3.3.2.3

Cancel the common factor.

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

Step 2.3.3.2.4

Rewrite the expression.

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

Step 2.3.3.2.5

Divide $x$ by $1$ .

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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 $3 (\frac{1}{3} y^{3})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Combine $3$ and $\frac{x^{2}}{2}$ .

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

Step 3.3

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

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

Step 3.4

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

$y = \sqrt[3]{\frac{3 x^{2}}{2} + \ln ({| x |}^{3}) + 3 C}$

Step 3.5

Simplify $\sqrt[3]{\frac{3 x^{2}}{2} + \ln ({| x |}^{3}) + 3 C}$ .

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Step 3.5.1

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

$y = \sqrt[3]{\frac{3 x^{2}}{2} + \ln ({| x |}^{3}) \cdot \frac{2}{2} + 3 C}$

Step 3.5.2

Simplify terms.

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Step 3.5.2.1

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

$y = \sqrt[3]{\frac{3 x^{2}}{2} + \frac{\ln ({| x |}^{3}) \cdot 2}{2} + 3 C}$

Step 3.5.2.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{3 x^{2} + \ln ({| x |}^{3}) \cdot 2}{2} + 3 C}$

Step 3.5.3

Simplify the numerator.

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Step 3.5.3.1

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

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Step 3.5.3.1.1

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

$y = \sqrt[3]{\frac{3 x^{2} + 2 \cdot \ln ({| x |}^{3})}{2} + 3 C}$

Step 3.5.3.1.2

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

$y = \sqrt[3]{\frac{3 x^{2} + \ln ({({| x |}^{3})}^{2})}{2} + 3 C}$

Step 3.5.3.2

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

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Step 3.5.3.2.1

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

$y = \sqrt[3]{\frac{3 x^{2} + \ln ({| x |}^{3 \cdot 2})}{2} + 3 C}$

Step 3.5.3.2.2

Multiply $3$ by $2$ .

$y = \sqrt[3]{\frac{3 x^{2} + \ln ({| x |}^{6})}{2} + 3 C}$

Step 3.5.3.3

Remove the absolute value in ${| x |}^{6}$ because exponentiations with even powers are always positive.

$y = \sqrt[3]{\frac{3 x^{2} + \ln (x^{6})}{2} + 3 C}$

Step 3.5.4

To write $3 C$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[3]{\frac{3 x^{2} + \ln (x^{6})}{2} + 3 C \cdot \frac{2}{2}}$

Step 3.5.5

Combine $3 C$ and $\frac{2}{2}$ .

$y = \sqrt[3]{\frac{3 x^{2} + \ln (x^{6})}{2} + \frac{3 C \cdot 2}{2}}$

Step 3.5.6

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{3 x^{2} + \ln (x^{6}) + 3 C \cdot 2}{2}}$

Step 3.5.7

Multiply $2$ by $3$ .

$y = \sqrt[3]{\frac{3 x^{2} + \ln (x^{6}) + 6 C}{2}}$

Step 3.5.8

Rewrite $\sqrt[3]{\frac{3 x^{2} + \ln (x^{6}) + 6 C}{2}}$ as $\frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C}}{\sqrt[3]{2}}$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C}}{\sqrt[3]{2}}$

Step 3.5.9

Multiply $\frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 3.5.10

Combine and simplify the denominator.

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Step 3.5.10.1

Multiply $\frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 3.5.10.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 3.5.10.3

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

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 3.5.10.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 3.5.10.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 3.5.10.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 3.5.10.5.2

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

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

Step 3.5.10.5.3

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

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.5.10.5.4

Cancel the common factor of $3$ .

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Step 3.5.10.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.5.10.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 3.5.10.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} {\sqrt[3]{2}}^{2}}{2}$

Step 3.5.11

Simplify the numerator.

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Step 3.5.11.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} \sqrt[3]{2^{2}}}{2}$

Step 3.5.11.2

Raise $2$ to the power of $2$ .

$y = \frac{\sqrt[3]{3 x^{2} + \ln (x^{6}) + 6 C} \sqrt[3]{4}}{2}$

Step 3.5.12

Simplify with factoring out.

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Step 3.5.12.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(3 x^{2} + \ln (x^{6}) + 6 C) \cdot 4}}{2}$

Step 3.5.12.2

Reorder factors in $\frac{\sqrt[3]{(3 x^{2} + \ln (x^{6}) + 6 C) \cdot 4}}{2}$ .

$y = \frac{\sqrt[3]{4 (3 x^{2} + \ln (x^{6}) + 6 C)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{4 (3 x^{2} + \ln (x^{6}) + C)}}{2}$