Calculus Examples

$1.45 = \frac{1}{2 \sqrt{64}} \cdot \frac{d x}{d t}$

Step 1

Separate the variables.

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Step 1.1

Flip sides to get $\frac{d x}{d t}$ on the left side.

$\frac{1}{2 \sqrt{64}} \frac{d x}{d t} = 1.45$

Step 1.2

Rewrite the equation.

$\frac{1}{2 \sqrt{64}} d x = 1.45 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{2 \sqrt{64}} d x = \int 1.45 d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply the constant rule.

$\frac{1}{2 \sqrt{64}} x + C_{1} = \int 1.45 d t$

Step 2.2.2

Rewrite $\frac{1}{2 \sqrt{64}} x + C_{1}$ as $\frac{1}{16} x + C_{1}$ .

$\frac{1}{16} x + C_{1} = \int 1.45 d t$

Step 2.3

Apply the constant rule.

$\frac{1}{16} x + C_{1} = 1.45 t + C_{2}$

Step 2.4

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

$\frac{1}{16} x = 1.45 t + K$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $16$ .

$16 (\frac{1}{16} x) = 16 (1.45 t + K)$

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 $16 (\frac{1}{16} x)$ .

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Step 3.2.1.1.1

Combine $\frac{1}{16}$ and $x$ .

$16 \frac{x}{16} = 16 (1.45 t + K)$

Step 3.2.1.1.2

Cancel the common factor of $16$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$16 \frac{x}{16} = 16 (1.45 t + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$x = 16 (1.45 t + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $16 (1.45 t + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$x = 16 (1.45 t) + 16 K$

Step 3.2.2.1.2

Multiply $1.45$ by $16$ .

$x = 23.2 t + 16 K$

Step 3.3

Reorder $23.2 t$ and $16 K$ .

$x = 16 K + 23.2 t$

Step 4

Simplify the constant of integration.

$x = K + 23.2 t$