Calculus Examples

$\frac{d L}{d t} = \frac{1}{6} \cdot \frac{d x}{d t}$

Step 1

Rewrite the equation.

$d L = \frac{1}{6} d L$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d L = \int \frac{1}{6} d L$

Step 2.2

Apply the constant rule.

$L + C_{1} = \int \frac{1}{6} d L$

Step 2.3

Apply the constant rule.

$L + C_{1} = \frac{1}{6} L + C_{2}$

Step 2.4

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

$L = \frac{1}{6} L + K$

Step 3

Solve for $L$ .

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Step 3.1

Combine $\frac{1}{6}$ and $L$ .

$L = \frac{L}{6} + K$

Step 3.2

Move all terms containing $L$ to the left side of the equation.

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Step 3.2.1

Subtract $\frac{L}{6}$ from both sides of the equation.

$L - \frac{L}{6} = K$

Step 3.2.2

To write $L$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$L \cdot \frac{6}{6} - \frac{L}{6} = K$

Step 3.2.3

Combine $L$ and $\frac{6}{6}$ .

$\frac{L \cdot 6}{6} - \frac{L}{6} = K$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{L \cdot 6 - L}{6} = K$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Move $6$ to the left of $L$ .

$\frac{6 \cdot L - L}{6} = K$

Step 3.2.5.2

Subtract $L$ from $6 L$ .

$\frac{5 L}{6} = K$

Step 3.3

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 L}{6} = \frac{6}{5} K$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $\frac{6}{5} \cdot \frac{5 L}{6}$ .

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Step 3.4.1.1.1

Cancel the common factor of $6$ .

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Step 3.4.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 L}{6} = \frac{6}{5} K$

Step 3.4.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 L) = \frac{6}{5} K$

Step 3.4.1.1.2

Cancel the common factor of $5$ .

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Step 3.4.1.1.2.1

Factor $5$ out of $5 L$ .

$\frac{1}{5} (5 (L)) = \frac{6}{5} K$

Step 3.4.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 L) = \frac{6}{5} K$

Step 3.4.1.1.2.3

Rewrite the expression.

$L = \frac{6}{5} K$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Combine $\frac{6}{5}$ and $K$ .

$L = \frac{6 K}{5}$

Step 4

Simplify the constant of integration.

$L = K$