Calculus Examples

$\frac{d y}{d x} = {(y + 2)}^{6}$ , $y (0) = - 1$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{{(y + 2)}^{6}}$ .

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

Step 1.2

Cancel the common factor of ${(y + 2)}^{6}$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = y + 2$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = y + 2$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y + 2$ .

$\frac{d}{d y} [y + 2]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $y + 2$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [2]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [2]$

Step 2.2.1.1.3

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

$1 + \frac{d}{d y} [2]$

Step 2.2.1.1.4

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

$\int \frac{1}{u^{6}} d u = \int d x$

Step 2.2.2

Apply basic rules of exponents.

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Step 2.2.2.1

Move $u^{6}$ out of the denominator by raising it to the $- 1$ power.

$\int {(u^{6})}^{- 1} d u = \int d x$

Step 2.2.2.2

Multiply the exponents in ${(u^{6})}^{- 1}$ .

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Step 2.2.2.2.1

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

$\int u^{6 \cdot - 1} d u = \int d x$

Step 2.2.2.2.2

Multiply $6$ by $- 1$ .

$\int u^{- 6} d u = \int d x$

Step 2.2.3

By the Power Rule, the integral of $u^{- 6}$ with respect to $u$ is $- \frac{1}{5} u^{- 5}$ .

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

Step 2.2.4

Simplify.

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Step 2.2.4.1

Rewrite $- \frac{1}{5} u^{- 5} + C_{1}$ as $- \frac{1}{5} \cdot \frac{1}{u^{5}} + C_{1}$ .

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

Step 2.2.4.2

Simplify.

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Step 2.2.4.2.1

Multiply $\frac{1}{u^{5}}$ by $\frac{1}{5}$ .

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

Step 2.2.4.2.2

Move $5$ to the left of $u^{5}$ .

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

Step 2.2.5

Replace all occurrences of $u$ with $y + 2$ .

$- \frac{1}{5 {(y + 2)}^{5}} + C_{1} = \int d x$

Step 2.3

Apply the constant rule.

$- \frac{1}{5 {(y + 2)}^{5}} + C_{1} = x + C_{2}$

Step 2.4

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

$- \frac{1}{5 {(y + 2)}^{5}} = x + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $- 1$ for $y$ in $- \frac{1}{5 {(y + 2)}^{5}} = x + K$ .

$- \frac{1}{5 {(- 1 + 2)}^{5}} = 0 + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $0 + K = - \frac{1}{5 {(- 1 + 2)}^{5}}$ .

$0 + K = - \frac{1}{5 {(- 1 + 2)}^{5}}$

Step 4.2

Add $0$ and $K$ .

$K = - \frac{1}{5 {(- 1 + 2)}^{5}}$

Step 4.3

Simplify $- \frac{1}{5 {(- 1 + 2)}^{5}}$ .

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Step 4.3.1

Simplify the denominator.

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Step 4.3.1.1

Add $- 1$ and $2$ .

$K = - \frac{1}{5 \cdot 1^{5}}$

Step 4.3.1.2

One to any power is one.

$K = - \frac{1}{5 \cdot 1}$

Step 4.3.2

Multiply $5$ by $1$ .

$K = - \frac{1}{5}$

Step 5

Substitute $- \frac{1}{5}$ for $K$ in $- \frac{1}{5 {(y + 2)}^{5}} = x + K$ and simplify.

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Step 5.1

Substitute $- \frac{1}{5}$ for $K$ .

$- \frac{1}{5 {(y + 2)}^{5}} = x - \frac{1}{5}$