Calculus Examples

$(5 - x) e^{y} d x = x y d y$

Step 1

Rewrite the equation.

$x y d y = (5 - x) e^{y} d x$

Step 2

Multiply both sides by $\frac{1}{x e^{y}}$ .

$\frac{1}{x e^{y}} (x y) d y = \frac{1}{x e^{y}} ((5 - x) e^{y}) d x$

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 $x e^{y}$ .

$\frac{1}{x (e^{y})} (x y) d y = \frac{1}{x e^{y}} ((5 - x) e^{y}) d x$

Step 3.1.2

Factor $x$ out of $x y$ .

$\frac{1}{x (e^{y})} (x (y)) d y = \frac{1}{x e^{y}} ((5 - x) e^{y}) d x$

Step 3.1.3

Cancel the common factor.

$\frac{1}{x e^{y}} (x y) d y = \frac{1}{x e^{y}} ((5 - x) e^{y}) d x$

Step 3.1.4

Rewrite the expression.

$\frac{1}{e^{y}} y d y = \frac{1}{x e^{y}} ((5 - x) e^{y}) d x$

Step 3.2

Combine $\frac{1}{e^{y}}$ and $y$ .

$\frac{y}{e^{y}} d y = \frac{1}{x e^{y}} ((5 - x) e^{y}) d x$

Step 3.3

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

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Step 3.3.1

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

$\frac{y}{e^{y}} d y = \frac{1}{e^{y} x} ((5 - x) e^{y}) d x$

Step 3.3.2

Factor $e^{y}$ out of $(5 - x) e^{y}$ .

$\frac{y}{e^{y}} d y = \frac{1}{e^{y} x} (e^{y} (5 - x)) d x$

Step 3.3.3

Cancel the common factor.

$\frac{y}{e^{y}} d y = \frac{1}{e^{y} x} (e^{y} (5 - x)) d x$

Step 3.3.4

Rewrite the expression.

$\frac{y}{e^{y}} d y = \frac{1}{x} (5 - x) d x$

Step 3.4

Multiply $\frac{1}{x}$ by $5 - x$ .

$\frac{y}{e^{y}} d y = \frac{5 - x}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{y}{e^{y}} d y = \int \frac{5 - x}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Simplify the expression.

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Step 4.2.1.1

Negate the exponent of $e^{y}$ and move it out of the denominator.

$\int y {(e^{y})}^{- 1} d y = \int \frac{5 - x}{x} d x$

Step 4.2.1.2

Multiply the exponents in ${(e^{y})}^{- 1}$ .

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Step 4.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int y e^{y \cdot - 1} d y = \int \frac{5 - x}{x} d x$

Step 4.2.1.2.2

Move $- 1$ to the left of $y$ .

$\int y e^{- 1 \cdot y} d y = \int \frac{5 - x}{x} d x$

Step 4.2.1.2.3

Rewrite $- 1 y$ as $- y$ .

$\int y e^{- y} d y = \int \frac{5 - x}{x} d x$

Step 4.2.2

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

$y (- e^{- y}) - \int - e^{- y} d y = \int \frac{5 - x}{x} d x$

Step 4.2.3

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

$y (- e^{- y}) - - \int e^{- y} d y = \int \frac{5 - x}{x} d x$

Step 4.2.4

Simplify.

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Step 4.2.4.1

Multiply $- 1$ by $- 1$ .

$y (- e^{- y}) + 1 \int e^{- y} d y = \int \frac{5 - x}{x} d x$

Step 4.2.4.2

Multiply $\int e^{- y} d y$ by $1$ .

$y (- e^{- y}) + \int e^{- y} d y = \int \frac{5 - x}{x} d x$

Step 4.2.5

Let $u = - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.2.5.1

Let $u = - y$ . Find $\frac{d u}{d y}$ .

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Step 4.2.5.1.1

Differentiate $- y$ .

$\frac{d}{d y} [- y]$

Step 4.2.5.1.2

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

$- \frac{d}{d y} [y]$

Step 4.2.5.1.3

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

$- 1 \cdot 1$

Step 4.2.5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.2.5.2

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

$y (- e^{- y}) + \int - e^{u} d u = \int \frac{5 - x}{x} d x$

Step 4.2.6

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

$y (- e^{- y}) - \int e^{u} d u = \int \frac{5 - x}{x} d x$

Step 4.2.7

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

$y (- e^{- y}) - (e^{u} + C_{1}) = \int \frac{5 - x}{x} d x$

Step 4.2.8

Rewrite $y (- e^{- y}) - (e^{u} + C_{1})$ as $- y e^{- y} - e^{u} + C_{1}$ .

$- y e^{- y} - e^{u} + C_{1} = \int \frac{5 - x}{x} d x$

Step 4.2.9

Replace all occurrences of $u$ with $- y$ .

$- y e^{- y} - e^{- y} + C_{1} = \int \frac{5 - x}{x} d x$

Step 4.2.10

Reorder terms.

$- e^{- y} y - e^{- y} + C_{1} = \int \frac{5 - x}{x} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

Split the fraction into multiple fractions.

$- e^{- y} y - e^{- y} + C_{1} = \int \frac{5}{x} + \frac{- x}{x} d x$

Step 4.3.2

Split the single integral into multiple integrals.

$- e^{- y} y - e^{- y} + C_{1} = \int \frac{5}{x} d x + \int \frac{- x}{x} d x$

Step 4.3.3

Cancel the common factor of $x$ .

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Step 4.3.3.1

Cancel the common factor.

$- e^{- y} y - e^{- y} + C_{1} = \int \frac{5}{x} d x + \int \frac{- x}{x} d x$

Step 4.3.3.2

Divide $- 1$ by $1$ .

$- e^{- y} y - e^{- y} + C_{1} = \int \frac{5}{x} d x + \int - 1 d x$

Step 4.3.4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$- e^{- y} y - e^{- y} + C_{1} = 5 \int \frac{1}{x} d x + \int - 1 d x$

Step 4.3.5

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

$- e^{- y} y - e^{- y} + C_{1} = 5 (\ln (| x |) + C_{2}) + \int - 1 d x$

Step 4.3.6

Apply the constant rule.

$- e^{- y} y - e^{- y} + C_{1} = 5 (\ln (| x |) + C_{2}) - x + C_{3}$

Step 4.3.7

Simplify.

$- e^{- y} y - e^{- y} + C_{1} = 5 \ln (| x |) - x + C_{4}$

Step 4.3.8

Reorder terms.

$- e^{- y} y - e^{- y} + C_{1} = - x + 5 \ln (| x |) + C_{4}$

Step 4.4

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

$- e^{- y} y - e^{- y} = - x + 5 \ln (| x |) + C$