Calculus Examples

$\frac{d y}{d x} = 9 e^{x + 7}$ , $y (- 7) = 0$

Step 1

Rewrite the equation.

$d y = 9 e^{x + 7} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 9 e^{x + 7} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 9 e^{x + 7} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$y + C_{1} = 9 \int e^{x + 7} d x$

Step 2.3.2

Let $u = x + 7$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = x + 7$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x + 7$ .

$\frac{d}{d x} [x + 7]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x + 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [7]$

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 $n = 1$ .

$1 + \frac{d}{d x} [7]$

Step 2.3.2.1.4

Since $7$ is constant with respect to $x$ , the derivative of $7$ 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 $d u$ .

$y + C_{1} = 9 \int e^{u} d u$

Step 2.3.3

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

$y + C_{1} = 9 (e^{u} + C_{2})$

Step 2.3.4

Simplify.

$y + C_{1} = 9 e^{u} + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $x + 7$ .

$y + C_{1} = 9 e^{x + 7} + C_{2}$

Step 2.4

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

$y = 9 e^{x + 7} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $- 7$ for $x$ and $0$ for $y$ in $y = 9 e^{x + 7} + K$ .

$0 = 9 e^{- 7 + 7} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $9 e^{- 7 + 7} + K = 0$ .

$9 e^{- 7 + 7} + K = 0$

Step 4.2

Simplify $9 e^{- 7 + 7} + K$ .

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Step 4.2.1

Add $- 7$ and $7$ .

$9 e^{0} + K = 0$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Anything raised to $0$ is $1$ .

$9 \cdot 1 + K = 0$

Step 4.2.2.2

Multiply $9$ by $1$ .

$9 + K = 0$

Step 4.3

Subtract $9$ from both sides of the equation.

$K = - 9$

Step 5

Substitute $- 9$ for $K$ in $y = 9 e^{x + 7} + K$ and simplify.

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Step 5.1

Substitute $- 9$ for $K$ .

$y = 9 e^{x + 7} - 9$