Calculus Examples

$\frac{d y}{d x} = x e^{- y} + 10 e^{- y}$

Step 1

Separate the variables.

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Step 1.1

Factor $e^{- y}$ out of $x e^{- y} + 10 e^{- y}$ .

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Step 1.1.1

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

$\frac{d y}{d x} = e^{- y} x + 10 e^{- y}$

Step 1.1.2

Factor $e^{- y}$ out of $10 e^{- y}$ .

$\frac{d y}{d x} = e^{- y} x + e^{- y} \cdot 10$

Step 1.1.3

Factor $e^{- y}$ out of $e^{- y} x + e^{- y} \cdot 10$ .

$\frac{d y}{d x} = e^{- y} (x + 10)$

Step 1.2

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

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{1}{e^{- y}} (e^{- y} (x + 10))$

Step 1.3

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

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Step 1.3.1

Cancel the common factor.

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{1}{e^{- y}} (e^{- y} (x + 10))$

Step 1.3.2

Rewrite the expression.

$\frac{1}{e^{- y}} \frac{d y}{d x} = x + 10$

Step 1.4

Rewrite the equation.

$\frac{1}{e^{- y}} d y = (x + 10) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{e^{- y}} d y = \int x + 10 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

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

$\int 1 {(e^{- y})}^{- 1} d y = \int x + 10 d x$

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

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

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Step 2.2.1.2.1.1

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

$\int 1 e^{- y \cdot - 1} d y = \int x + 10 d x$

Step 2.2.1.2.1.2

Multiply $- y \cdot - 1$ .

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Step 2.2.1.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\int 1 e^{1 y} d y = \int x + 10 d x$

Step 2.2.1.2.1.2.2

Multiply $y$ by $1$ .

$\int 1 e^{y} d y = \int x + 10 d x$

Step 2.2.1.2.2

Multiply $e^{y}$ by $1$ .

$\int e^{y} d y = \int x + 10 d x$

Step 2.2.2

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

$e^{y} + C_{1} = \int x + 10 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$e^{y} + C_{1} = \int x d x + \int 10 d x$

Step 2.3.2

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

$e^{y} + C_{1} = \frac{1}{2} x^{2} + C_{2} + \int 10 d x$

Step 2.3.3

Apply the constant rule.

$e^{y} + C_{1} = \frac{1}{2} x^{2} + C_{2} + 10 x + C_{3}$

Step 2.3.4

Simplify.

$e^{y} + C_{1} = \frac{1}{2} x^{2} + 10 x + C_{4}$

Step 2.4

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

$e^{y} = \frac{1}{2} x^{2} + 10 x + K$

Step 3

Solve for $y$ .

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Step 3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y}) = \ln (\frac{x^{2}}{2} + 10 x + K)$

Step 3.2

Expand the left side.

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Step 3.2.1

Expand $\ln (e^{y})$ by moving $y$ outside the logarithm.

$y \ln (e) = \ln (\frac{x^{2}}{2} + 10 x + K)$

Step 3.2.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (\frac{x^{2}}{2} + 10 x + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (\frac{x^{2}}{2} + 10 x + K)$