Calculus Examples

$d x + e^{- 5 x} d y = 0$

Step 1

Subtract $d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $\frac{1}{e^{- 5 x}}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $e^{- 5 x}$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Combine $\frac{1}{e^{- 5 x}}$ and $- 1$ .

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

Step 3.3

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Apply the constant rule.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Negate the exponent of $e^{- 5 x}$ and move it out of the denominator.

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

Step 4.3.2.2

Simplify.

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Step 4.3.2.2.1

Multiply the exponents in ${(e^{- 5 x})}^{- 1}$ .

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Step 4.3.2.2.1.1

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

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

Step 4.3.2.2.1.2

Multiply $- 1$ by $- 5$ .

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

Step 4.3.2.2.2

Multiply $e^{5 x}$ by $1$ .

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

Step 4.3.3

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.3.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 4.3.3.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 4.3.3.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 4.3.3.1.3

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

$5 \cdot 1$

Step 4.3.3.1.4

Multiply $5$ by $1$ .

$5$

Step 4.3.3.2

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

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

Step 4.3.4

Combine $e^{u}$ and $\frac{1}{5}$ .

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

Step 4.3.5

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

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

Step 4.3.6

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

$y + C_{1} = - \frac{1}{5} (e^{u} + C_{2})$

Step 4.3.7

Simplify.

$y + C_{1} = - \frac{1}{5} e^{u} + C_{2}$

Step 4.3.8

Replace all occurrences of $u$ with $5 x$ .

$y + C_{1} = - \frac{1}{5} e^{5 x} + C_{2}$

Step 4.4

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

$y = - \frac{1}{5} e^{5 x} + K$