Calculus Examples

$x e^{y} d y + (\frac{x^{2} + 1}{y}) d x = 0$

Step 1

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

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

Step 2

Multiply both sides by $\frac{y}{x}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x e^{y}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

Move the leading negative in $- \frac{y}{x}$ into the numerator.

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

Step 3.3.2

Factor $y$ out of $- y$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Move the negative in front of the fraction.

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Cancel the common factor of $x$ .

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Step 3.6.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.6.2

Factor $x$ out of $x^{2}$ .

$y e^{y} d y = (\frac{- 1}{x} (x \cdot x) - \frac{1}{x} \cdot 1) d x$

Step 3.6.3

Cancel the common factor.

$y e^{y} d y = (\frac{- 1}{x} (x \cdot x) - \frac{1}{x} \cdot 1) d x$

Step 3.6.4

Rewrite the expression.

$y e^{y} d y = (- x - \frac{1}{x} \cdot 1) d x$

Step 3.7

Multiply $- 1$ by $1$ .

$y e^{y} d y = (- x - \frac{1}{x}) d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Simplify.

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

Step 4.2.4

Reorder terms.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Split the single integral into multiple integrals.

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 4.3.6

Simplify.

$e^{y} y - e^{y} + C_{1} = - \frac{1}{2} x^{2} - \ln (| x |) + C_{4}$

Step 4.4

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

$e^{y} y - e^{y} = - \frac{1}{2} x^{2} - \ln (| x |) + C$