Calculus Examples

$\frac{d y}{d x} = \frac{x - 4}{\sqrt{x^{2} - 8 x + 2}}$

Step 1

Rewrite the equation.

$d y = \frac{x - 4}{\sqrt{x^{2} - 8 x + 2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{x - 4}{\sqrt{x^{2} - 8 x + 2}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{x - 4}{\sqrt{x^{2} - 8 x + 2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = x^{2} - 8 x + 2$ . Then $d u = (2 x - 8) d x$ , so $\frac{1}{2} d u = (x - 4) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = x^{2} - 8 x + 2$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $x^{2} - 8 x + 2$ .

$\frac{d}{d x} [x^{2} - 8 x + 2]$

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 8 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [2]$

Step 2.3.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [2]$

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.3.1.1.3.2

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

$2 x - 8 \cdot 1 + \frac{d}{d x} [2]$

Step 2.3.1.1.3.3

Multiply $- 8$ by $1$ .

$2 x - 8 + \frac{d}{d x} [2]$

Step 2.3.1.1.4

Differentiate using the Constant Rule.

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Step 2.3.1.1.4.1

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

$2 x - 8 + 0$

Step 2.3.1.1.4.2

Add $2 x - 8$ and $0$ .

$2 x - 8$

Step 2.3.1.2

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

$y + C_{1} = \int \frac{1}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{2}$ .

$y + C_{1} = \int \frac{1}{\sqrt{u} \cdot 2} d u$

Step 2.3.2.2

Move $2$ to the left of $\sqrt{u}$ .

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

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.4

Apply basic rules of exponents.

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Step 2.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 2.3.4.2

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

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

Step 2.3.4.3

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

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Step 2.3.4.3.1

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

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

Step 2.3.4.3.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 2.3.4.3.3

Move the negative in front of the fraction.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

$y + C_{1} = \frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C_{2}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.3.6.2.2

Cancel the common factor of $2$ .

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Step 2.3.6.2.2.1

Cancel the common factor.

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

Step 2.3.6.2.2.2

Rewrite the expression.

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

Step 2.3.6.2.3

Multiply $u^{\frac{1}{2}}$ by $1$ .

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

Step 2.3.7

Replace all occurrences of $u$ with $x^{2} - 8 x + 2$ .

$y + C_{1} = {(x^{2} - 8 x + 2)}^{\frac{1}{2}} + C_{2}$

Step 2.4

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

$y = {(x^{2} - 8 x + 2)}^{\frac{1}{2}} + K$