Calculus Examples

$\frac{d p}{d t} = N (1 - \frac{N}{K})$

Step 1

Rewrite the equation.

$d p = N (1 - \frac{N}{K}) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d p = \int N (1 - \frac{N}{K}) d t$

Step 2.2

Apply the constant rule.

$p + C_{1} = \int N (1 - \frac{N}{K}) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply the constant rule.

$p + C_{1} = N (1 - \frac{N}{K}) t + C_{2}$

Step 2.3.2

Simplify the answer.

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Step 2.3.2.1

Rewrite $N (1 - \frac{N}{K}) t + C_{2}$ as $(N - N \frac{N}{K}) t + C_{2}$ .

$p + C_{1} = (N - N \frac{N}{K}) t + C_{2}$

Step 2.3.2.2

Simplify.

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Step 2.3.2.2.1

Combine $\frac{N}{K}$ and $N$ .

$p + C_{1} = (N - \frac{N \cdot N}{K}) t + C_{2}$

Step 2.3.2.2.2

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

$p + C_{1} = (N - \frac{N^{1} N}{K}) t + C_{2}$

Step 2.3.2.2.3

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

$p + C_{1} = (N - \frac{N^{1} N^{1}}{K}) t + C_{2}$

Step 2.3.2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p + C_{1} = (N - \frac{N^{1 + 1}}{K}) t + C_{2}$

Step 2.3.2.2.5

Add $1$ and $1$ .

$p + C_{1} = (N - \frac{N^{2}}{K}) t + C_{2}$

Step 2.3.3

Reorder terms.

$p + C_{1} = t (N - \frac{N^{2}}{K}) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K_{1}$ .

$p = t (N - \frac{N^{2}}{K}) + K_{1}$