Calculus Examples

$- \frac{9}{15} \cdot \frac{d θ}{d t} = \frac{1}{15} (18)$

Step 1

Rewrite the equation.

$- \frac{9}{15} d θ = \frac{1}{15} (18) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int - \frac{9}{15} d θ = \int \frac{1}{15} \cdot 18 d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Cancel the common factor of $9$ and $15$ .

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Step 2.2.1.1

Factor $3$ out of $9$ .

$\int - \frac{3 (3)}{15} d θ = \int \frac{1}{15} \cdot 18 d t$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $3$ out of $15$ .

$\int - \frac{3 \cdot 3}{3 \cdot 5} d θ = \int \frac{1}{15} \cdot 18 d t$

Step 2.2.1.2.2

Cancel the common factor.

$\int - \frac{3 \cdot 3}{3 \cdot 5} d θ = \int \frac{1}{15} \cdot 18 d t$

Step 2.2.1.2.3

Rewrite the expression.

$\int - \frac{3}{5} d θ = \int \frac{1}{15} \cdot 18 d t$

Step 2.2.2

Apply the constant rule.

$- \frac{3}{5} θ + C_{1} = \int \frac{1}{15} \cdot 18 d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify.

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Step 2.3.1.1

Combine $\frac{1}{15}$ and $18$ .

$- \frac{3}{5} θ + C_{1} = \int \frac{18}{15} d t$

Step 2.3.1.2

Cancel the common factor of $18$ and $15$ .

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Step 2.3.1.2.1

Factor $3$ out of $18$ .

$- \frac{3}{5} θ + C_{1} = \int \frac{3 (6)}{15} d t$

Step 2.3.1.2.2

Cancel the common factors.

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Step 2.3.1.2.2.1

Factor $3$ out of $15$ .

$- \frac{3}{5} θ + C_{1} = \int \frac{3 \cdot 6}{3 \cdot 5} d t$

Step 2.3.1.2.2.2

Cancel the common factor.

$- \frac{3}{5} θ + C_{1} = \int \frac{3 \cdot 6}{3 \cdot 5} d t$

Step 2.3.1.2.2.3

Rewrite the expression.

$- \frac{3}{5} θ + C_{1} = \int \frac{6}{5} d t$

Step 2.3.2

Apply the constant rule.

$- \frac{3}{5} θ + C_{1} = \frac{6}{5} t + C_{2}$

Step 2.4

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

$- \frac{3}{5} θ = \frac{6}{5} t + K$

Step 3

Solve for $θ$ .

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Step 3.1

Multiply both sides of the equation by $- \frac{5}{3}$ .

$- \frac{5}{3} (- \frac{3}{5} θ) = - \frac{5}{3} (\frac{6}{5} t + K)$

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 $- \frac{5}{3} (- \frac{3}{5} θ)$ .

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Step 3.2.1.1.1

Combine $θ$ and $\frac{3}{5}$ .

$- \frac{5}{3} (- \frac{θ \cdot 3}{5}) = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.2.1.1.2.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$\frac{- 5}{3} (- \frac{θ \cdot 3}{5}) = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.2.2

Move the leading negative in $- \frac{θ \cdot 3}{5}$ into the numerator.

$\frac{- 5}{3} \cdot \frac{- θ \cdot 3}{5} = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.2.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{3} \cdot \frac{- θ \cdot 3}{5} = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.2.4

Cancel the common factor.

$\frac{5 \cdot - 1}{3} \cdot \frac{- θ \cdot 3}{5} = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.2.5

Rewrite the expression.

$\frac{- 1}{3} (- θ \cdot 3) = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.3

Cancel the common factor of $3$ .

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Step 3.2.1.1.3.1

Factor $3$ out of $- θ \cdot 3$ .

$\frac{- 1}{3} (3 \cdot (- θ)) = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.3.2

Cancel the common factor.

$\frac{- 1}{3} (3 \cdot (- θ)) = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.3.3

Rewrite the expression.

$- - θ = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.4

Multiply.

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Step 3.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 θ = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.1.1.4.2

Multiply $θ$ by $1$ .

$θ = - \frac{5}{3} (\frac{6}{5} t + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{5}{3} (\frac{6}{5} t + K)$ .

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Step 3.2.2.1.1

Simplify terms.

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Step 3.2.2.1.1.1

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

$θ = - \frac{5}{3} (\frac{6 t}{5} + K)$

Step 3.2.2.1.1.2

Apply the distributive property.

$θ = - \frac{5}{3} \cdot \frac{6 t}{5} - \frac{5}{3} K$

Step 3.2.2.1.1.3

Cancel the common factor of $5$ .

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Step 3.2.2.1.1.3.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$θ = \frac{- 5}{3} \cdot \frac{6 t}{5} - \frac{5}{3} K$

Step 3.2.2.1.1.3.2

Factor $5$ out of $- 5$ .

$θ = \frac{5 (- 1)}{3} \cdot \frac{6 t}{5} - \frac{5}{3} K$

Step 3.2.2.1.1.3.3

Cancel the common factor.

$θ = \frac{5 \cdot - 1}{3} \cdot \frac{6 t}{5} - \frac{5}{3} K$

Step 3.2.2.1.1.3.4

Rewrite the expression.

$θ = \frac{- 1}{3} (6 t) - \frac{5}{3} K$

Step 3.2.2.1.1.4

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.4.1

Factor $3$ out of $6 t$ .

$θ = \frac{- 1}{3} (3 (2 t)) - \frac{5}{3} K$

Step 3.2.2.1.1.4.2

Cancel the common factor.

$θ = \frac{- 1}{3} (3 (2 t)) - \frac{5}{3} K$

Step 3.2.2.1.1.4.3

Rewrite the expression.

$θ = - (2 t) - \frac{5}{3} K$

Step 3.2.2.1.1.5

Multiply $2$ by $- 1$ .

$θ = - 2 t - \frac{5}{3} K$

Step 3.2.2.1.1.6

Combine $K$ and $\frac{5}{3}$ .

$θ = - 2 t - \frac{K \cdot 5}{3}$

Step 3.2.2.1.2

Move $5$ to the left of $K$ .

$θ = - 2 t - \frac{5 K}{3}$

Step 3.3

Reorder $- 2 t$ and $- \frac{5 K}{3}$ .

$θ = - \frac{5 K}{3} - 2 t$

Step 4

Simplify the constant of integration.

$θ = K - 2 t$