Calculus Examples

$x^{2} \frac{d w}{d x} = \sqrt{w} (8 x + 1)$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (8 x + 1)$ by $x^{2}$ and simplify.

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Step 1.1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (8 x + 1)$ by $x^{2}$ .

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (8 x + 1)}{x^{2}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x^{2}$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (8 x + 1)}{x^{2}}$

Step 1.1.2.1.2

Divide $\frac{d w}{d x}$ by $1$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (8 x + 1)}{x^{2}}$

Step 1.2

Regroup factors.

$\frac{d w}{d x} = \sqrt{w} \frac{8 x + 1}{x^{2}}$

Step 1.3

Multiply both sides by $\frac{1}{\sqrt{w}}$ .

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{1}{\sqrt{w}} (\sqrt{w} \frac{8 x + 1}{x^{2}})$

Step 1.4

Cancel the common factor of $\sqrt{w}$ .

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Step 1.4.1

Cancel the common factor.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{1}{\sqrt{w}} (\sqrt{w} \frac{8 x + 1}{x^{2}})$

Step 1.4.2

Rewrite the expression.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{8 x + 1}{x^{2}}$

Step 1.5

Rewrite the equation.

$\frac{1}{\sqrt{w}} d w = \frac{8 x + 1}{x^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{w}} d w = \int \frac{8 x + 1}{x^{2}} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

$\int \frac{1}{w^{\frac{1}{2}}} d w = \int \frac{8 x + 1}{x^{2}} d x$

Step 2.2.1.2

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

$\int {(w^{\frac{1}{2}})}^{- 1} d w = \int \frac{8 x + 1}{x^{2}} d x$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$\int w^{\frac{1}{2} \cdot - 1} d w = \int \frac{8 x + 1}{x^{2}} d x$

Step 2.2.1.3.2

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

$\int w^{\frac{- 1}{2}} d w = \int \frac{8 x + 1}{x^{2}} d x$

Step 2.2.1.3.3

Move the negative in front of the fraction.

$\int w^{- \frac{1}{2}} d w = \int \frac{8 x + 1}{x^{2}} d x$

Step 2.2.2

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

$2 w^{\frac{1}{2}} + C_{1} = \int \frac{8 x + 1}{x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

$2 w^{\frac{1}{2}} + C_{1} = \int (8 x + 1) x^{2 \cdot - 1} d x$

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2

Multiply $(8 x + 1) x^{- 2}$ .

$2 w^{\frac{1}{2}} + C_{1} = \int 8 x \cdot x^{- 2} + 1 x^{- 2} d x$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 2.3.3.1.1

Move $x^{- 2}$ .

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

Step 2.3.3.1.2

Multiply $x^{- 2}$ by $x$ .

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Step 2.3.3.1.2.1

Raise $x$ to the power of $1$ .

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

Step 2.3.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.1.3

Add $- 2$ and $1$ .

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

Step 2.3.3.2

Multiply $x^{- 2}$ by $1$ .

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

Step 2.3.4

Split the single integral into multiple integrals.

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

Step 2.3.5

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

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

Step 2.3.6

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

$2 w^{\frac{1}{2}} + C_{1} = 8 (\ln (| x |) + C_{2}) + \int x^{- 2} d x$

Step 2.3.7

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

$2 w^{\frac{1}{2}} + C_{1} = 8 (\ln (| x |) + C_{2}) - x^{- 1} + C_{3}$

Step 2.3.8

Simplify.

$2 w^{\frac{1}{2}} + C_{1} = 8 \ln (| x |) - \frac{1}{x} + C_{4}$

Step 2.3.9

Reorder terms.

$2 w^{\frac{1}{2}} + C_{1} = - \frac{1}{x} + 8 \ln (| x |) + C_{4}$

Step 2.4

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

$2 w^{\frac{1}{2}} = - \frac{1}{x} + 8 \ln (| x |) + C$

Step 3

Solve for $w$ .

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Step 3.1

Divide each term in $2 w^{\frac{1}{2}} = - \frac{1}{x} + 8 \ln (| x |) + C$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 w^{\frac{1}{2}} = - \frac{1}{x} + 8 \ln (| x |) + C$ by $2$ .

$\frac{2 w^{\frac{1}{2}}}{2} = \frac{- \frac{1}{x}}{2} + \frac{8 \ln (| x |)}{2} + \frac{C}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

$\frac{2 w^{\frac{1}{2}}}{2} = \frac{- \frac{1}{x}}{2} + \frac{8 \ln (| x |)}{2} + \frac{C}{2}$

Step 3.1.2.2

Divide $w^{\frac{1}{2}}$ by $1$ .

$w^{\frac{1}{2}} = \frac{- \frac{1}{x}}{2} + \frac{8 \ln (| x |)}{2} + \frac{C}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$w^{\frac{1}{2}} = - \frac{1}{x} \cdot \frac{1}{2} + \frac{8 \ln (| x |)}{2} + \frac{C}{2}$

Step 3.1.3.1.2

Multiply $\frac{1}{2}$ by $\frac{1}{x}$ .

$w^{\frac{1}{2}} = - \frac{1}{2 x} + \frac{8 \ln (| x |)}{2} + \frac{C}{2}$

Step 3.1.3.1.3

Cancel the common factor of $8$ and $2$ .

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Step 3.1.3.1.3.1

Factor $2$ out of $8 \ln (| x |)$ .

$w^{\frac{1}{2}} = - \frac{1}{2 x} + \frac{2 (4 \ln (| x |))}{2} + \frac{C}{2}$

Step 3.1.3.1.3.2

Cancel the common factors.

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Step 3.1.3.1.3.2.1

Factor $2$ out of $2$ .

$w^{\frac{1}{2}} = - \frac{1}{2 x} + \frac{2 (4 \ln (| x |))}{2 (1)} + \frac{C}{2}$

Step 3.1.3.1.3.2.2

Cancel the common factor.

$w^{\frac{1}{2}} = - \frac{1}{2 x} + \frac{2 (4 \ln (| x |))}{2 \cdot 1} + \frac{C}{2}$

Step 3.1.3.1.3.2.3

Rewrite the expression.

$w^{\frac{1}{2}} = - \frac{1}{2 x} + \frac{4 \ln (| x |)}{1} + \frac{C}{2}$

Step 3.1.3.1.3.2.4

Divide $4 \ln (| x |)$ by $1$ .

$w^{\frac{1}{2}} = - \frac{1}{2 x} + 4 \ln (| x |) + \frac{C}{2}$

Step 3.1.3.1.4

Simplify $4 \ln (| x |)$ by moving $4$ inside the logarithm.

$w^{\frac{1}{2}} = - \frac{1}{2 x} + \ln ({| x |}^{4}) + \frac{C}{2}$

Step 3.1.3.1.5

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

$w^{\frac{1}{2}} = - \frac{1}{2 x} + \ln (x^{4}) + \frac{C}{2}$

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(w^{\frac{1}{2}})}^{2} = {(- \frac{1}{2 x} + \ln (x^{4}) + \frac{C}{2})}^{2}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${(w^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

$w^{\frac{1}{2} \cdot 2} = {(- \frac{1}{2 x} + \ln (x^{4}) + \frac{C}{2})}^{2}$

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$w^{\frac{1}{2} \cdot 2} = {(- \frac{1}{2 x} + \ln (x^{4}) + \frac{C}{2})}^{2}$

Step 3.3.1.1.2.2

Rewrite the expression.

$w^{1} = {(- \frac{1}{2 x} + \ln (x^{4}) + \frac{C}{2})}^{2}$

Step 3.3.1.2

Simplify.

$w = {(- \frac{1}{2 x} + \ln (x^{4}) + \frac{C}{2})}^{2}$

Step 4

Simplify the constant of integration.

$w = {(- \frac{1}{2 x} + \ln (x^{4}) + C)}^{2}$