Calculus Examples

$\frac{d N}{d t} N = N t e^{t + 2}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $\frac{d N}{d t} N = N t e^{t + 2}$ by $N$ and simplify.

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Step 1.1.1

Divide each term in $\frac{d N}{d t} N = N t e^{t + 2}$ by $N$ .

$\frac{\frac{d N}{d t} N}{N} = \frac{N t e^{t + 2}}{N}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $N$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{\frac{d N}{d t} N}{N} = \frac{N t e^{t + 2}}{N}$

Step 1.1.2.1.2

Divide $\frac{d N}{d t}$ by $1$ .

$\frac{d N}{d t} = \frac{N t e^{t + 2}}{N}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Cancel the common factor of $N$ .

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Step 1.1.3.1.1

Cancel the common factor.

$\frac{d N}{d t} = \frac{N t e^{t + 2}}{N}$

Step 1.1.3.1.2

Divide $t e^{t + 2}$ by $1$ .

$\frac{d N}{d t} = t e^{t + 2}$

Step 1.2

Rewrite the equation.

$d N = t e^{t + 2} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d N = \int t e^{t + 2} d t$

Step 2.2

Apply the constant rule.

$N + C_{1} = \int t e^{t + 2} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = e^{t + 2}$ .

$N + C_{1} = t e^{t + 2} - \int e^{t + 2} d t$

Step 2.3.2

Let $u = t + 2$ . Then $d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = t + 2$ . Find $\frac{d u}{d t}$ .

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Step 2.3.2.1.1

Differentiate $t + 2$ .

$\frac{d}{d t} [t + 2]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $t + 2$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [2]$ .

$\frac{d}{d t} [t] + \frac{d}{d t} [2]$

Step 2.3.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d t} [2]$

Step 2.3.2.1.4

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$1 + 0$

Step 2.3.2.1.5

Add $1$ and $0$ .

$1$

Step 2.3.2.2

Rewrite the problem using $u$ and $d u$ .

$N + C_{1} = t e^{t + 2} - \int e^{u} d u$

Step 2.3.3

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$N + C_{1} = t e^{t + 2} - (e^{u} + C_{2})$

Step 2.3.4

Simplify.

$N + C_{1} = t e^{t + 2} - e^{u} + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $t + 2$ .

$N + C_{1} = t e^{t + 2} - e^{t + 2} + C_{2}$

Step 2.3.6

Reorder terms.

$N + C_{1} = e^{t + 2} t - e^{t + 2} + C_{2}$

Step 2.4

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

$N = e^{t + 2} t - e^{t + 2} + K$