Calculus Examples

$35 \frac{d y}{d t} = - 56$

Step 1

Rewrite the equation.

$35 d y = - 56 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 35 d y = \int - 56 d t$

Step 2.2

Apply the constant rule.

$35 y + C_{1} = \int - 56 d t$

Step 2.3

Apply the constant rule.

$35 y + C_{1} = - 56 t + C_{2}$

Step 2.4

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

$35 y = - 56 t + K$

Step 3

Divide each term in $35 y = - 56 t + K$ by $35$ and simplify.

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Step 3.1

Divide each term in $35 y = - 56 t + K$ by $35$ .

$\frac{35 y}{35} = \frac{- 56 t}{35} + \frac{K}{35}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $35$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{35 y}{35} = \frac{- 56 t}{35} + \frac{K}{35}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 56 t}{35} + \frac{K}{35}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $- 56$ and $35$ .

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Step 3.3.1.1.1

Factor $7$ out of $- 56 t$ .

$y = \frac{7 (- 8 t)}{35} + \frac{K}{35}$

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

Factor $7$ out of $35$ .

$y = \frac{7 (- 8 t)}{7 (5)} + \frac{K}{35}$

Step 3.3.1.1.2.2

Cancel the common factor.

$y = \frac{7 (- 8 t)}{7 \cdot 5} + \frac{K}{35}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{- 8 t}{5} + \frac{K}{35}$

Step 3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{8 t}{5} + \frac{K}{35}$

Step 4

Simplify the constant of integration.

$y = - \frac{8 t}{5} + K$