Calculus Examples

$\frac{d y}{d x} + x e^{- \frac{x^{2}}{2}} = 0$

Step 1

Separate the variables.

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Step 1.1

Subtract $x e^{- \frac{x^{2}}{2}}$ from both sides of the equation.

$\frac{d y}{d x} = - x e^{- \frac{x^{2}}{2}}$

Step 1.2

Rewrite the equation.

$d y = - x e^{- \frac{x^{2}}{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - x e^{- \frac{x^{2}}{2}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - x e^{- \frac{x^{2}}{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = - \int x e^{- \frac{x^{2}}{2}} d x$

Step 2.3.2

Let $u_{2} = e^{- \frac{x^{2}}{2}}$ . Then $d u_{2} = - x e^{- \frac{x^{2}}{2}} d x$ , so $\frac{1}{- x e^{- \frac{x^{2}}{2}}} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.2.1

Let $u_{2} = e^{- \frac{x^{2}}{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $e^{- \frac{x^{2}}{2}}$ .

$\frac{d}{d x} [e^{- \frac{x^{2}}{2}}]$

Step 2.3.2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \frac{x^{2}}{2}$ .

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Step 2.3.2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- \frac{x^{2}}{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- \frac{x^{2}}{2}]$

Step 2.3.2.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- \frac{x^{2}}{2}]$

Step 2.3.2.1.2.3

Replace all occurrences of $u_{1}$ with $- \frac{x^{2}}{2}$ .

$e^{- \frac{x^{2}}{2}} \frac{d}{d x} [- \frac{x^{2}}{2}]$

Step 2.3.2.1.3

Differentiate.

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Step 2.3.2.1.3.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$e^{- \frac{x^{2}}{2}} (- \frac{1}{2} \frac{d}{d x} [x^{2}])$

Step 2.3.2.1.3.2

Combine $e^{- \frac{x^{2}}{2}}$ and $\frac{1}{2}$ .

$- \frac{e^{- \frac{x^{2}}{2}}}{2} \frac{d}{d x} [x^{2}]$

Step 2.3.2.1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- \frac{e^{- \frac{x^{2}}{2}}}{2} (2 x)$

Step 2.3.2.1.3.4

Simplify terms.

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Step 2.3.2.1.3.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{e^{- \frac{x^{2}}{2}}}{2} x$

Step 2.3.2.1.3.4.2

Combine $- 2$ and $\frac{e^{- \frac{x^{2}}{2}}}{2}$ .

$\frac{- 2 e^{- \frac{x^{2}}{2}}}{2} x$

Step 2.3.2.1.3.4.3

Combine $\frac{- 2 e^{- \frac{x^{2}}{2}}}{2}$ and $x$ .

$\frac{- 2 e^{- \frac{x^{2}}{2}} x}{2}$

Step 2.3.2.1.3.4.4

Cancel the common factor of $- 2$ and $2$ .

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Step 2.3.2.1.3.4.4.1

Factor $2$ out of $- 2 e^{- \frac{x^{2}}{2}} x$ .

$\frac{2 (- e^{- \frac{x^{2}}{2}} x)}{2}$

Step 2.3.2.1.3.4.4.2

Cancel the common factors.

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Step 2.3.2.1.3.4.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (- e^{- \frac{x^{2}}{2}} x)}{2 (1)}$

Step 2.3.2.1.3.4.4.2.2

Cancel the common factor.

$\frac{2 (- e^{- \frac{x^{2}}{2}} x)}{2 \cdot 1}$

Step 2.3.2.1.3.4.4.2.3

Rewrite the expression.

$\frac{- e^{- \frac{x^{2}}{2}} x}{1}$

Step 2.3.2.1.3.4.4.2.4

Divide $- e^{- \frac{x^{2}}{2}} x$ by $1$ .

$- e^{- \frac{x^{2}}{2}} x$

Step 2.3.2.1.3.4.5

Reorder factors in $- e^{- \frac{x^{2}}{2}} x$ .

$- x e^{- \frac{x^{2}}{2}}$

Step 2.3.2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$y + C_{1} = - \int - 1 d u_{2}$

Step 2.3.3

Apply the constant rule.

$y + C_{1} = - (- u_{2} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Simplify.

$y + C_{1} = - - u_{2} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $- 1$ by $- 1$ .

$y + C_{1} = 1 u_{2} + C_{2}$

Step 2.3.4.2.2

Multiply $u_{2}$ by $1$ .

$y + C_{1} = u_{2} + C_{2}$

Step 2.3.4.3

Replace all occurrences of $u_{2}$ with $e^{- \frac{x^{2}}{2}}$ .

$y + C_{1} = e^{- \frac{x^{2}}{2}} + C_{2}$

Step 2.3.4.4

Reorder terms.

$y + C_{1} = e^{- \frac{1}{2} x^{2}} + C_{2}$

Step 2.4

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

$y = e^{- \frac{1}{2} x^{2}} + K$