Calculus Examples

$x \sqrt{x^{2} + 1} - y e^{y} \frac{d y}{d x} = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Reorder factors in $x \sqrt{x^{2} + 1} - y e^{y} \frac{d y}{d x}$ .

$x \sqrt{x^{2} + 1} - y \frac{d y}{d x} e^{y} = 0$

Step 1.1.2

Subtract $x \sqrt{x^{2} + 1}$ from both sides of the equation.

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

Step 1.1.3

Divide each term in $- y \frac{d y}{d x} e^{y} = - x \sqrt{x^{2} + 1}$ by $- y e^{y}$ and simplify.

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Step 1.1.3.1

Divide each term in $- y \frac{d y}{d x} e^{y} = - x \sqrt{x^{2} + 1}$ by $- y e^{y}$ .

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.3.2.2

Cancel the common factor of $y$ .

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Step 1.1.3.2.2.1

Cancel the common factor.

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

Step 1.1.3.2.2.2

Rewrite the expression.

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

Step 1.1.3.2.3

Cancel the common factor of $e^{y}$ .

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Step 1.1.3.2.3.1

Cancel the common factor.

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

Step 1.1.3.2.3.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Dividing two negative values results in a positive value.

$\frac{d y}{d x} = \frac{x \sqrt{x^{2} + 1}}{y e^{y}}$

Step 1.2

Multiply both sides by $y e^{y}$ .

$y e^{y} \frac{d y}{d x} = y e^{y} \frac{x \sqrt{x^{2} + 1}}{y e^{y}}$

Step 1.3

Cancel the common factor of $y e^{y}$ .

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Step 1.3.1

Cancel the common factor.

$y e^{y} \frac{d y}{d x} = y e^{y} \frac{x \sqrt{x^{2} + 1}}{y e^{y}}$

Step 1.3.2

Rewrite the expression.

$y e^{y} \frac{d y}{d x} = x \sqrt{x^{2} + 1}$

Step 1.4

Rewrite the equation.

$y e^{y} d y = x \sqrt{x^{2} + 1} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y e^{y} d y = \int x \sqrt{x^{2} + 1} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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

Step 2.2.4

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = x^{2} + 1$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 2.3.1.1.3

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

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

Step 2.3.1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.3.1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.1.2

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

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

Step 2.3.2

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

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

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 2.3.5

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$e^{y} y - e^{y} + C_{1} = \frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + C_{2})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Rewrite $\frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + C_{2})$ as $\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C_{2}$ .

$e^{y} y - e^{y} + C_{1} = \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$e^{y} y - e^{y} + C_{1} = \frac{2}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.6.2.2

Multiply $2$ by $3$ .

$e^{y} y - e^{y} + C_{1} = \frac{2}{6} u^{\frac{3}{2}} + C_{2}$

Step 2.3.6.2.3

Cancel the common factor of $2$ and $6$ .

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Step 2.3.6.2.3.1

Factor $2$ out of $2$ .

$e^{y} y - e^{y} + C_{1} = \frac{2 (1)}{6} u^{\frac{3}{2}} + C_{2}$

Step 2.3.6.2.3.2

Cancel the common factors.

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Step 2.3.6.2.3.2.1

Factor $2$ out of $6$ .

$e^{y} y - e^{y} + C_{1} = \frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.6.2.3.2.2

Cancel the common factor.

$e^{y} y - e^{y} + C_{1} = \frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.6.2.3.2.3

Rewrite the expression.

$e^{y} y - e^{y} + C_{1} = \frac{1}{3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7

Replace all occurrences of $u$ with $x^{2} + 1$ .

$e^{y} y - e^{y} + C_{1} = \frac{1}{3} {(x^{2} + 1)}^{\frac{3}{2}} + C_{2}$

Step 2.4

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

$e^{y} y - e^{y} = \frac{1}{3} {(x^{2} + 1)}^{\frac{3}{2}} + K$