Calculus Examples

$\frac{d y}{d x} = - 4 e^{x - 8}$ , $y (8) = 8$

Step 1

Rewrite the equation.

$d y = - 4 e^{x - 8} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 4 e^{x - 8} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 4 e^{x - 8} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$y + C_{1} = - 4 \int e^{x - 8} d x$

Step 2.3.2

Let $u = x - 8$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = x - 8$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x - 8$ .

$\frac{d}{d x} [x - 8]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x - 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$

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 $n = 1$ .

$1 + \frac{d}{d x} [- 8]$

Step 2.3.2.1.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.2.1.5

Add $1$ and $0$ .

$1$

Step 2.3.2.2

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

$y + C_{1} = - 4 \int e^{u} d u$

Step 2.3.3

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

$y + C_{1} = - 4 (e^{u} + C_{2})$

Step 2.3.4

Simplify.

$y + C_{1} = - 4 e^{u} + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $x - 8$ .

$y + C_{1} = - 4 e^{x - 8} + C_{2}$

Step 2.4

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

$y = - 4 e^{x - 8} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $8$ for $x$ and $8$ for $y$ in $y = - 4 e^{x - 8} + K$ .

$8 = - 4 e^{8 - 8} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- 4 e^{8 - 8} + K = 8$ .

$- 4 e^{8 - 8} + K = 8$

Step 4.2

Simplify $- 4 e^{8 - 8} + K$ .

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Step 4.2.1

Subtract $8$ from $8$ .

$- 4 e^{0} + K = 8$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Anything raised to $0$ is $1$ .

$- 4 \cdot 1 + K = 8$

Step 4.2.2.2

Multiply $- 4$ by $1$ .

$- 4 + K = 8$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Add $4$ to both sides of the equation.

$K = 8 + 4$

Step 4.3.2

Add $8$ and $4$ .

$K = 12$

Step 5

Substitute $12$ for $K$ in $y = - 4 e^{x - 8} + K$ and simplify.

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Step 5.1

Substitute $12$ for $K$ .

$y = - 4 e^{x - 8} + 12$