Calculus Examples

$\frac{d y}{d x} = \frac{1}{19 y^{18}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y^{18}$ .

$y^{18} \frac{d y}{d x} = y^{18} \frac{1}{19 y^{18}}$

Step 1.2

Cancel the common factor of $y^{18}$ .

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Step 1.2.1

Factor $y^{18}$ out of $19 y^{18}$ .

$y^{18} \frac{d y}{d x} = y^{18} \frac{1}{y^{18} \cdot 19}$

Step 1.2.2

Cancel the common factor.

$y^{18} \frac{d y}{d x} = y^{18} \frac{1}{y^{18} \cdot 19}$

Step 1.2.3

Rewrite the expression.

$y^{18} \frac{d y}{d x} = \frac{1}{19}$

Step 1.3

Rewrite the equation.

$y^{18} d y = \frac{1}{19} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{18} d y = \int \frac{1}{19} d x$

Step 2.2

By the Power Rule, the integral of $y^{18}$ with respect to $y$ is $\frac{1}{19} y^{19}$ .

$\frac{1}{19} y^{19} + C_{1} = \int \frac{1}{19} d x$

Step 2.3

Apply the constant rule.

$\frac{1}{19} y^{19} + C_{1} = \frac{1}{19} x + C_{2}$

Step 2.4

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

$\frac{1}{19} y^{19} = \frac{1}{19} x + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $19$ .

$19 (\frac{1}{19} y^{19}) = 19 (\frac{1}{19} x + K)$

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 $19 (\frac{1}{19} y^{19})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{19}$ and $y^{19}$ .

$19 \frac{y^{19}}{19} = 19 (\frac{1}{19} x + K)$

Step 3.2.1.1.2

Cancel the common factor of $19$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$19 \frac{y^{19}}{19} = 19 (\frac{1}{19} x + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{19} = 19 (\frac{1}{19} x + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $19 (\frac{1}{19} x + K)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{19}$ and $x$ .

$y^{19} = 19 (\frac{x}{19} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{19} = 19 \frac{x}{19} + 19 K$

Step 3.2.2.1.3

Cancel the common factor of $19$ .

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Step 3.2.2.1.3.1

Cancel the common factor.

$y^{19} = 19 \frac{x}{19} + 19 K$

Step 3.2.2.1.3.2

Rewrite the expression.

$y^{19} = x + 19 K$

Step 3.3

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

$y = \sqrt[19]{x + 19 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt[19]{x + K}$