Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $\frac{1}{6} (\frac{1}{3} x^{3} + C_{2})$ as $\frac{1}{6} \cdot \frac{1}{3} x^{3} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Multiply $\frac{1}{6}$ by $\frac{1}{3}$ .

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

Step 2.3.3.2.2

Multiply $6$ by $3$ .

$e^{y} + C_{1} = \frac{1}{18} x^{3} + C_{2}$

Step 2.4

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

$e^{y} = \frac{1}{18} x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y}) = \ln (\frac{x^{3}}{18} + K)$

Step 3.2

Expand the left side.

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Step 3.2.1

Expand $\ln (e^{y})$ by moving $y$ outside the logarithm.

$y \ln (e) = \ln (\frac{x^{3}}{18} + K)$

Step 3.2.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (\frac{x^{3}}{18} + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (\frac{x^{3}}{18} + K)$