Calculus Examples

$\frac{d y}{d x} = \frac{4 x^{3} + e^{x}}{5 y^{4}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y^{4}$ .

$y^{4} \frac{d y}{d x} = y^{4} \frac{4 x^{3} + e^{x}}{5 y^{4}}$

Step 1.2

Cancel the common factor of $y^{4}$ .

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Step 1.2.1

Factor $y^{4}$ out of $5 y^{4}$ .

$y^{4} \frac{d y}{d x} = y^{4} \frac{4 x^{3} + e^{x}}{y^{4} \cdot 5}$

Step 1.2.2

Cancel the common factor.

$y^{4} \frac{d y}{d x} = y^{4} \frac{4 x^{3} + e^{x}}{y^{4} \cdot 5}$

Step 1.2.3

Rewrite the expression.

$y^{4} \frac{d y}{d x} = \frac{4 x^{3} + e^{x}}{5}$

Step 1.3

Rewrite the equation.

$y^{4} d y = \frac{4 x^{3} + e^{x}}{5} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{4} d y = \int \frac{4 x^{3} + e^{x}}{5} d x$

Step 2.2

By the Power Rule, the integral of $y^{4}$ with respect to $y$ is $\frac{1}{5} y^{5}$ .

$\frac{1}{5} y^{5} + C_{1} = \int \frac{4 x^{3} + e^{x}}{5} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} y^{5} + C_{1} = \frac{1}{5} \int 4 x^{3} + e^{x} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$\frac{1}{5} y^{5} + C_{1} = \frac{1}{5} (\int 4 x^{3} d x + \int e^{x} d x)$

Step 2.3.3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{5} y^{5} + C_{1} = \frac{1}{5} (4 \int x^{3} d x + \int e^{x} d x)$

Step 2.3.4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{5} y^{5} + C_{1} = \frac{1}{5} (4 (\frac{1}{4} x^{4} + C_{2}) + \int e^{x} d x)$

Step 2.3.5

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

$\frac{1}{5} y^{5} + C_{1} = \frac{1}{5} (4 (\frac{1}{4} x^{4} + C_{2}) + e^{x} + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{1}{5} y^{5} + C_{1} = \frac{1}{5} (4 (\frac{x^{4}}{4} + C_{2}) + e^{x} + C_{3})$

Step 2.3.6.2

Simplify.

$\frac{1}{5} y^{5} + C_{1} = \frac{1}{5} (x^{4} + e^{x}) + C_{4}$

Step 2.4

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

$\frac{1}{5} y^{5} = \frac{1}{5} (x^{4} + e^{x}) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $5$ .

$5 (\frac{1}{5} y^{5}) = 5 (\frac{1}{5} (x^{4} + e^{x}) + K)$

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 $5 (\frac{1}{5} y^{5})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{5}$ and $y^{5}$ .

$5 \frac{y^{5}}{5} = 5 (\frac{1}{5} (x^{4} + e^{x}) + K)$

Step 3.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$5 \frac{y^{5}}{5} = 5 (\frac{1}{5} (x^{4} + e^{x}) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{5} = 5 (\frac{1}{5} (x^{4} + e^{x}) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $5 (\frac{1}{5} (x^{4} + e^{x}) + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$y^{5} = 5 (\frac{1}{5} x^{4} + \frac{1}{5} e^{x} + K)$

Step 3.2.2.1.1.2

Combine $\frac{1}{5}$ and $x^{4}$ .

$y^{5} = 5 (\frac{x^{4}}{5} + \frac{1}{5} e^{x} + K)$

Step 3.2.2.1.1.3

Combine $\frac{1}{5}$ and $e^{x}$ .

$y^{5} = 5 (\frac{x^{4}}{5} + \frac{e^{x}}{5} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{5} = 5 \frac{x^{4}}{5} + 5 \frac{e^{x}}{5} + 5 K$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.3.1.1

Cancel the common factor.

$y^{5} = 5 \frac{x^{4}}{5} + 5 \frac{e^{x}}{5} + 5 K$

Step 3.2.2.1.3.1.2

Rewrite the expression.

$y^{5} = x^{4} + 5 \frac{e^{x}}{5} + 5 K$

Step 3.2.2.1.3.2

Cancel the common factor of $5$ .

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Step 3.2.2.1.3.2.1

Cancel the common factor.

$y^{5} = x^{4} + 5 \frac{e^{x}}{5} + 5 K$

Step 3.2.2.1.3.2.2

Rewrite the expression.

$y^{5} = x^{4} + e^{x} + 5 K$

Step 3.3

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

$y = \sqrt[5]{x^{4} + e^{x} + 5 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt[5]{x^{4} + e^{x} + K}$