Calculus Examples

$\frac{d y}{d x} = 25 e^{- 5 x}$

Step 1

Rewrite the equation.

$d y = 25 e^{- 5 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 25 e^{- 5 x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 25 e^{- 5 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $25$ is constant with respect to $x$ , move $25$ out of the integral.

$y + C_{1} = 25 \int e^{- 5 x} d x$

Step 2.3.2

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 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $- 5 x$ .

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

Step 2.3.2.1.2

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 2.3.2.1.3

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 2.3.2.1.4

Multiply $- 5$ by $1$ .

$- 5$

Step 2.3.2.2

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

$y + C_{1} = 25 \int e^{u} \frac{1}{- 5} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Move the negative in front of the fraction.

$y + C_{1} = 25 \int e^{u} (- \frac{1}{5}) d u$

Step 2.3.3.2

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

$y + C_{1} = 25 \int - \frac{e^{u}}{5} d u$

Step 2.3.4

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

$y + C_{1} = 25 (- \int \frac{e^{u}}{5} d u)$

Step 2.3.5

Multiply $- 1$ by $25$ .

$y + C_{1} = - 25 \int \frac{e^{u}}{5} d u$

Step 2.3.6

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

$y + C_{1} = - 25 (\frac{1}{5} \int e^{u} d u)$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Combine $\frac{1}{5}$ and $- 25$ .

$y + C_{1} = \frac{- 25}{5} \int e^{u} d u$

Step 2.3.7.2

Cancel the common factor of $- 25$ and $5$ .

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Step 2.3.7.2.1

Factor $5$ out of $- 25$ .

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

Step 2.3.7.2.2

Cancel the common factors.

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Step 2.3.7.2.2.1

Factor $5$ out of $5$ .

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

Step 2.3.7.2.2.2

Cancel the common factor.

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

Step 2.3.7.2.2.3

Rewrite the expression.

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

Step 2.3.7.2.2.4

Divide $- 5$ by $1$ .

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

Step 2.3.8

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

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

Step 2.3.9

Simplify.

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

Step 2.3.10

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

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

Step 2.4

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

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