Calculus Examples

$\frac{d y}{d x} = \frac{8}{1 + x}$ , $y (0) = 7$

Step 1

Rewrite the equation.

$d y = \frac{8}{1 + x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{8}{1 + x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{8}{1 + x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$y + C_{1} = 8 \int \frac{1}{1 + x} d x$

Step 2.3.2

Let $u = 1 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 1 + x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $1 + x$ .

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

Step 2.3.2.1.2

By the Sum Rule, the derivative of $1 + x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x]$ .

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

Step 2.3.2.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x]$

Step 2.3.2.1.4

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

$0 + 1$

Step 2.3.2.1.5

Add $0$ and $1$ .

$1$

Step 2.3.2.2

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

$y + C_{1} = 8 \int \frac{1}{u} d u$

Step 2.3.3

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$y + C_{1} = 8 (\ln (| u |) + C_{2})$

Step 2.3.4

Simplify.

$y + C_{1} = 8 \ln (| u |) + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $1 + x$ .

$y + C_{1} = 8 \ln (| 1 + x |) + C_{2}$

Step 2.4

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

$y = 8 \ln (| 1 + x |) + C$

Step 3

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $7$ for $y$ in $y = 8 \ln (| 1 + x |) + C$ .

$7 = 8 \ln (| 1 + 0 |) + C$

Step 4

Solve for $C$ .

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Step 4.1

Rewrite the equation as $8 \ln (| 1 + 0 |) + C = 7$ .

$8 \ln (| 1 + 0 |) + C = 7$

Step 4.2

Simplify $8 \ln (| 1 + 0 |) + C$ .

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Step 4.2.1

Add $1$ and $0$ .

$8 \ln (| 1 |) + C = 7$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$8 \ln (1) + C = 7$

Step 4.2.2.2

The natural logarithm of $1$ is $0$ .

$8 \cdot 0 + C = 7$

Step 4.2.2.3

Multiply $8$ by $0$ .

$0 + C = 7$

Step 4.2.3

Add $0$ and $C$ .

$C = 7$

Step 5

Substitute $7$ for $C$ in $y = 8 \ln (| 1 + x |) + C$ and simplify.

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Step 5.1

Substitute $7$ for $C$ .

$y = 8 \ln (| 1 + x |) + 7$

Step 5.2

Simplify each term.

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Step 5.2.1

Simplify $8 \ln (| 1 + x |)$ by moving $8$ inside the logarithm.

$y = \ln ({| 1 + x |}^{8}) + 7$

Step 5.2.2

Remove the absolute value in ${| 1 + x |}^{8}$ because exponentiations with even powers are always positive.

$y = \ln ({(1 + x)}^{8}) + 7$