Calculus Examples

$\frac{d v}{d x} = \frac{s + 1}{s v + s}$

Step 1

Separate the variables.

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Step 1.1

Factor $s$ out of $s v + s$ .

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Step 1.1.1

Factor $s$ out of $s v$ .

$\frac{d v}{d x} = \frac{s + 1}{s (v) + s}$

Step 1.1.2

Raise $s$ to the power of $1$ .

$\frac{d v}{d x} = \frac{s + 1}{s (v) + s^{1}}$

Step 1.1.3

Factor $s$ out of $s^{1}$ .

$\frac{d v}{d x} = \frac{s + 1}{s (v) + s \cdot 1}$

Step 1.1.4

Factor $s$ out of $s (v) + s \cdot 1$ .

$\frac{d v}{d x} = \frac{s + 1}{s (v + 1)}$

Step 1.2

Regroup factors.

$\frac{d v}{d x} = \frac{s + 1}{s} \cdot \frac{1}{v + 1}$

Step 1.3

Multiply both sides by $v + 1$ .

$(v + 1) \frac{d v}{d x} = (v + 1) (\frac{s + 1}{s} \cdot \frac{1}{v + 1})$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{s + 1}{s}$ by $\frac{1}{v + 1}$ .

$(v + 1) \frac{d v}{d x} = (v + 1) \frac{s + 1}{s (v + 1)}$

Step 1.4.2

Cancel the common factor of $v + 1$ .

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Step 1.4.2.1

Factor $v + 1$ out of $s (v + 1)$ .

$(v + 1) \frac{d v}{d x} = (v + 1) \frac{s + 1}{(v + 1) s}$

Step 1.4.2.2

Cancel the common factor.

$(v + 1) \frac{d v}{d x} = (v + 1) \frac{s + 1}{(v + 1) s}$

Step 1.4.2.3

Rewrite the expression.

$(v + 1) \frac{d v}{d x} = \frac{s + 1}{s}$

Step 1.5

Rewrite the equation.

$(v + 1) d v = \frac{s + 1}{s} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int v + 1 d v = \int \frac{s + 1}{s} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int v d v + \int d v = \int \frac{s + 1}{s} d x$

Step 2.2.2

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

$\frac{1}{2} v^{2} + C_{1} + \int d v = \int \frac{s + 1}{s} d x$

Step 2.2.3

Apply the constant rule.

$\frac{1}{2} v^{2} + C_{1} + v + C_{2} = \int \frac{s + 1}{s} d x$

Step 2.2.4

Simplify.

$\frac{1}{2} v^{2} + v + C_{3} = \int \frac{s + 1}{s} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

$\frac{1}{2} v^{2} + v + C_{3} = \int \frac{s}{s} + \frac{1}{s} d x$

Step 2.3.2

Apply the constant rule.

$\frac{1}{2} v^{2} + v + C_{3} = (\frac{s}{s} + \frac{1}{s}) x + C_{4}$

Step 2.3.3

Cancel the common factor of $s$ .

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Step 2.3.3.1

Cancel the common factor.

$\frac{1}{2} v^{2} + v + C_{3} = (\frac{s}{s} + \frac{1}{s}) x + C_{4}$

Step 2.3.3.2

Rewrite the expression.

$\frac{1}{2} v^{2} + v + C_{3} = (1 + \frac{1}{s}) x + C_{4}$

Step 2.3.4

Reorder terms.

$\frac{1}{2} v^{2} + v + C_{3} = x (1 + \frac{1}{s}) + C_{4}$

Step 2.4

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

$\frac{1}{2} v^{2} + v = x (1 + \frac{1}{s}) + K$