Calculus Examples

$\frac{d y}{d x} = \frac{x (e^{x^{2}} + 2)}{6 y^{2}}$

Step 1

Separate the variables.

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Step 1.1

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

$y^{2} \frac{d y}{d x} = y^{2} \frac{x (e^{x^{2}} + 2)}{6 y^{2}}$

Step 1.2

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

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Step 1.2.1

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

$y^{2} \frac{d y}{d x} = y^{2} \frac{x (e^{x^{2}} + 2)}{y^{2} \cdot 6}$

Step 1.2.2

Cancel the common factor.

$y^{2} \frac{d y}{d x} = y^{2} \frac{x (e^{x^{2}} + 2)}{y^{2} \cdot 6}$

Step 1.2.3

Rewrite the expression.

$y^{2} \frac{d y}{d x} = \frac{x (e^{x^{2}} + 2)}{6}$

Step 1.3

Rewrite the equation.

$y^{2} d y = \frac{x (e^{x^{2}} + 2)}{6} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int \frac{x (e^{x^{2}} + 2)}{6} d x$

Step 2.2

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

$\frac{1}{3} y^{3} + C_{1} = \int \frac{x (e^{x^{2}} + 2)}{6} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{6} \int x (e^{x^{2}} + 2) d x$

Step 2.3.2

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

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Step 2.3.2.1

Let $u = x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 2.3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 2.3.2.2

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

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{6} \int (e^{u} + 2) \frac{1}{2} d u$

Step 2.3.3

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

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{6} (\frac{1}{2} \int e^{u} + 2 d u)$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Multiply $\frac{1}{2}$ by $\frac{1}{6}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{2 \cdot 6} \int e^{u} + 2 d u$

Step 2.3.4.2

Multiply $2$ by $6$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} \int e^{u} + 2 d u$

Step 2.3.5

Split the single integral into multiple integrals.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} (\int e^{u} d u + \int 2 d u)$

Step 2.3.6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} (e^{u} + C_{2} + \int 2 d u)$

Step 2.3.7

Apply the constant rule.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} (e^{u} + C_{2} + 2 u + C_{3})$

Step 2.3.8

Simplify.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} (e^{u} + 2 u) + C_{4}$

Step 2.3.9

Replace all occurrences of $u$ with $x^{2}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} (e^{x^{2}} + 2 x^{2}) + C_{4}$

Step 2.3.10

Simplify.

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Step 2.3.10.1

Apply the distributive property.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} e^{x^{2}} + \frac{1}{12} (2 x^{2}) + C_{4}$

Step 2.3.10.2

Combine $\frac{1}{12}$ and $e^{x^{2}}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{e^{x^{2}}}{12} + \frac{1}{12} (2 x^{2}) + C_{4}$

Step 2.3.10.3

Cancel the common factor of $2$ .

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Step 2.3.10.3.1

Factor $2$ out of $12$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{e^{x^{2}}}{12} + \frac{1}{2 (6)} (2 x^{2}) + C_{4}$

Step 2.3.10.3.2

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

$\frac{1}{3} y^{3} + C_{1} = \frac{e^{x^{2}}}{12} + \frac{1}{2 (6)} (2 (x^{2})) + C_{4}$

Step 2.3.10.3.3

Cancel the common factor.

$\frac{1}{3} y^{3} + C_{1} = \frac{e^{x^{2}}}{12} + \frac{1}{2 \cdot 6} (2 x^{2}) + C_{4}$

Step 2.3.10.3.4

Rewrite the expression.

$\frac{1}{3} y^{3} + C_{1} = \frac{e^{x^{2}}}{12} + \frac{1}{6} x^{2} + C_{4}$

Step 2.3.10.4

Combine $\frac{1}{6}$ and $x^{2}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{e^{x^{2}}}{12} + \frac{x^{2}}{6} + C_{4}$

Step 2.3.11

Reorder terms.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{12} e^{x^{2}} + \frac{1}{6} x^{2} + C_{4}$

Step 2.4

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

$\frac{1}{3} y^{3} = \frac{1}{12} e^{x^{2}} + \frac{1}{6} x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\frac{1}{12} e^{x^{2}} + \frac{1}{6} x^{2} + K)$

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 $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (\frac{1}{12} e^{x^{2}} + \frac{1}{6} x^{2} + K)$

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.

$3 \frac{y^{3}}{3} = 3 (\frac{1}{12} e^{x^{2}} + \frac{1}{6} x^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\frac{1}{12} e^{x^{2}} + \frac{1}{6} x^{2} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (\frac{1}{12} e^{x^{2}} + \frac{1}{6} x^{2} + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Combine $\frac{1}{12}$ and $e^{x^{2}}$ .

$y^{3} = 3 (\frac{e^{x^{2}}}{12} + \frac{1}{6} x^{2} + K)$

Step 3.2.2.1.1.2

Combine $\frac{1}{6}$ and $x^{2}$ .

$y^{3} = 3 (\frac{e^{x^{2}}}{12} + \frac{x^{2}}{6} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = 3 \frac{e^{x^{2}}}{12} + 3 \frac{x^{2}}{6} + 3 K$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.1.1

Factor $3$ out of $12$ .

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

Step 3.2.2.1.3.1.2

Cancel the common factor.

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

Step 3.2.2.1.3.1.3

Rewrite the expression.

$y^{3} = \frac{e^{x^{2}}}{4} + 3 \frac{x^{2}}{6} + 3 K$

Step 3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.2.1

Factor $3$ out of $6$ .

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

Step 3.2.2.1.3.2.2

Cancel the common factor.

$y^{3} = \frac{e^{x^{2}}}{4} + 3 \frac{x^{2}}{3 \cdot 2} + 3 K$

Step 3.2.2.1.3.2.3

Rewrite the expression.

$y^{3} = \frac{e^{x^{2}}}{4} + \frac{x^{2}}{2} + 3 K$

Step 3.3

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

$y = \sqrt[3]{\frac{e^{x^{2}}}{4} + \frac{x^{2}}{2} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{\frac{e^{x^{2}}}{4} + \frac{x^{2}}{2} + 3 K}$ .

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Step 3.4.1

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

$y = \sqrt[3]{\frac{e^{x^{2}}}{4} + \frac{x^{2}}{2} \cdot \frac{2}{2} + 3 K}$

Step 3.4.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.1

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

$y = \sqrt[3]{\frac{e^{x^{2}}}{4} + \frac{x^{2} \cdot 2}{2 \cdot 2} + 3 K}$

Step 3.4.2.2

Multiply $2$ by $2$ .

$y = \sqrt[3]{\frac{e^{x^{2}}}{4} + \frac{x^{2} \cdot 2}{4} + 3 K}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{e^{x^{2}} + x^{2} \cdot 2}{4} + 3 K}$

Step 3.4.4

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

$y = \sqrt[3]{\frac{e^{x^{2}} + 2 x^{2}}{4} + 3 K}$

Step 3.4.5

To write $3 K$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \sqrt[3]{\frac{e^{x^{2}} + 2 x^{2}}{4} + 3 K \cdot \frac{4}{4}}$

Step 3.4.6

Combine $3 K$ and $\frac{4}{4}$ .

$y = \sqrt[3]{\frac{e^{x^{2}} + 2 x^{2}}{4} + \frac{3 K \cdot 4}{4}}$

Step 3.4.7

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{e^{x^{2}} + 2 x^{2} + 3 K \cdot 4}{4}}$

Step 3.4.8

Multiply $4$ by $3$ .

$y = \sqrt[3]{\frac{e^{x^{2}} + 2 x^{2} + 12 K}{4}}$

Step 3.4.9

Rewrite $\sqrt[3]{\frac{e^{x^{2}} + 2 x^{2} + 12 K}{4}}$ as $\frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K}}{\sqrt[3]{4}}$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K}}{\sqrt[3]{4}}$

Step 3.4.10

Multiply $\frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 3.4.11

Combine and simplify the denominator.

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Step 3.4.11.1

Multiply $\frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 3.4.11.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.4.11.3

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

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.4.11.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 3.4.11.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 3.4.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.4.11.5.2

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

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.4.11.5.3

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

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.11.5.4

Cancel the common factor of $3$ .

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Step 3.4.11.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.11.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.4.11.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} {\sqrt[3]{4}}^{2}}{4}$

Step 3.4.12

Simplify the numerator.

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Step 3.4.12.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} \sqrt[3]{4^{2}}}{4}$

Step 3.4.12.2

Raise $4$ to the power of $2$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} \sqrt[3]{16}}{4}$

Step 3.4.12.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 3.4.12.3.1

Factor $8$ out of $16$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} \sqrt[3]{8 (2)}}{4}$

Step 3.4.12.3.2

Rewrite $8$ as $2^{3}$ .

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 3.4.12.4

Pull terms out from under the radical.

$y = \frac{\sqrt[3]{e^{x^{2}} + 2 x^{2} + 12 K} \cdot 2 \sqrt[3]{2}}{4}$

Step 3.4.12.5

Combine using the product rule for radicals.

$y = \frac{2 \sqrt[3]{(e^{x^{2}} + 2 x^{2} + 12 K) \cdot 2}}{4}$

Step 3.4.13

Reduce the expression by cancelling the common factors.

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Step 3.4.13.1

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

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Step 3.4.13.1.1

Factor $2$ out of $2 \sqrt[3]{(e^{x^{2}} + 2 x^{2} + 12 K) \cdot 2}$ .

$y = \frac{2 \sqrt[3]{(e^{x^{2}} + 2 x^{2} + 12 K) \cdot 2}}{4}$

Step 3.4.13.1.2

Cancel the common factors.

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Step 3.4.13.1.2.1

Factor $2$ out of $4$ .

$y = \frac{2 \sqrt[3]{(e^{x^{2}} + 2 x^{2} + 12 K) \cdot 2}}{2 \cdot 2}$

Step 3.4.13.1.2.2

Cancel the common factor.

$y = \frac{2 \sqrt[3]{(e^{x^{2}} + 2 x^{2} + 12 K) \cdot 2}}{2 \cdot 2}$

Step 3.4.13.1.2.3

Rewrite the expression.

$y = \frac{\sqrt[3]{(e^{x^{2}} + 2 x^{2} + 12 K) \cdot 2}}{2}$

Step 3.4.13.2

Reorder factors in $\frac{\sqrt[3]{(e^{x^{2}} + 2 x^{2} + 12 K) \cdot 2}}{2}$ .

$y = \frac{\sqrt[3]{2 (e^{x^{2}} + 2 x^{2} + 12 K)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{2 (e^{x^{2}} + 2 x^{2} + K)}}{2}$