Calculus Examples

$\frac{d y}{d x} = \frac{7}{{(6 + x)}^{2}}$ , $y (0) = 6$

Step 1

Rewrite the equation.

$d y = \frac{7}{{(6 + x)}^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{7}{{(6 + x)}^{2}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{7}{{(6 + x)}^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$y + C_{1} = 7 \int \frac{1}{{(6 + x)}^{2}} d x$

Step 2.3.2

Let $u = 6 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 6 + x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $6 + x$ .

$\frac{d}{d x} [6 + x]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $6 + x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [x]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [x]$

Step 2.3.2.1.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ 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 $d u$ .

$y + C_{1} = 7 \int \frac{1}{u^{2}} d u$

Step 2.3.3

Apply basic rules of exponents.

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Step 2.3.3.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

$y + C_{1} = 7 \int {(u^{2})}^{- 1} d u$

Step 2.3.3.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 2.3.3.2.1

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

$y + C_{1} = 7 \int u^{2 \cdot - 1} d u$

Step 2.3.3.2.2

Multiply $2$ by $- 1$ .

$y + C_{1} = 7 \int u^{- 2} d u$

Step 2.3.4

By the Power Rule, the integral of $u^{- 2}$ with respect to $u$ is $- u^{- 1}$ .

$y + C_{1} = 7 (- u^{- 1} + C_{2})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Rewrite $7 (- u^{- 1} + C_{2})$ as $7 (- \frac{1}{u}) + C_{2}$ .

$y + C_{1} = 7 (- \frac{1}{u}) + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Multiply $- 1$ by $7$ .

$y + C_{1} = - 7 \frac{1}{u} + C_{2}$

Step 2.3.5.2.2

Combine $- 7$ and $\frac{1}{u}$ .

$y + C_{1} = \frac{- 7}{u} + C_{2}$

Step 2.3.5.2.3

Move the negative in front of the fraction.

$y + C_{1} = - \frac{7}{u} + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $6 + x$ .

$y + C_{1} = - \frac{7}{6 + x} + C_{2}$

Step 2.4

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

$y = - \frac{7}{6 + x} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $6$ for $y$ in $y = - \frac{7}{6 + x} + K$ .

$6 = - \frac{7}{6 + 0} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{7}{6 + 0} + K = 6$ .

$- \frac{7}{6 + 0} + K = 6$

Step 4.2

Add $6$ and $0$ .

$- \frac{7}{6} + K = 6$

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 $\frac{7}{6}$ to both sides of the equation.

$K = 6 + \frac{7}{6}$

Step 4.3.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$K = 6 \cdot \frac{6}{6} + \frac{7}{6}$

Step 4.3.3

Combine $6$ and $\frac{6}{6}$ .

$K = \frac{6 \cdot 6}{6} + \frac{7}{6}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{6 \cdot 6 + 7}{6}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $6$ by $6$ .

$K = \frac{36 + 7}{6}$

Step 4.3.5.2

Add $36$ and $7$ .

$K = \frac{43}{6}$

Step 5

Substitute $\frac{43}{6}$ for $K$ in $y = - \frac{7}{6 + x} + K$ and simplify.

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Step 5.1

Substitute $\frac{43}{6}$ for $K$ .

$y = - \frac{7}{6 + x} + \frac{43}{6}$

Step 5.2

To write $- \frac{7}{6 + x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = - \frac{7}{6 + x} \cdot \frac{6}{6} + \frac{43}{6}$

Step 5.3

To write $\frac{43}{6}$ as a fraction with a common denominator, multiply by $\frac{6 + x}{6 + x}$ .

$y = - \frac{7}{6 + x} \cdot \frac{6}{6} + \frac{43}{6} \cdot \frac{6 + x}{6 + x}$

Step 5.4

Write each expression with a common denominator of $(6 + x) \cdot 6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{7}{6 + x}$ by $\frac{6}{6}$ .

$y = - \frac{7 \cdot 6}{(6 + x) \cdot 6} + \frac{43}{6} \cdot \frac{6 + x}{6 + x}$

Step 5.4.2

Multiply $\frac{43}{6}$ by $\frac{6 + x}{6 + x}$ .

$y = - \frac{7 \cdot 6}{(6 + x) \cdot 6} + \frac{43 (6 + x)}{6 (6 + x)}$

Step 5.4.3

Reorder the factors of $(6 + x) \cdot 6$ .

$y = - \frac{7 \cdot 6}{6 (6 + x)} + \frac{43 (6 + x)}{6 (6 + x)}$

Step 5.5

Combine the numerators over the common denominator.

$y = \frac{- 7 \cdot 6 + 43 (6 + x)}{6 (6 + x)}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $- 7$ by $6$ .

$y = \frac{- 42 + 43 (6 + x)}{6 (6 + x)}$

Step 5.6.2

Apply the distributive property.

$y = \frac{- 42 + 43 \cdot 6 + 43 x}{6 (6 + x)}$

Step 5.6.3

Multiply $43$ by $6$ .

$y = \frac{- 42 + 258 + 43 x}{6 (6 + x)}$

Step 5.6.4

Add $- 42$ and $258$ .

$y = \frac{216 + 43 x}{6 (6 + x)}$