Calculus Examples

$\frac{d y}{d x} = y^{- 2} \ln (x)$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{- 2}}$ .

$\frac{1}{y^{- 2}} \frac{d y}{d x} = \frac{1}{y^{- 2}} (y^{- 2} \ln (x))$

Step 1.2

Cancel the common factor of $y^{- 2}$ .

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Step 1.2.1

Factor $y^{- 2}$ out of $y^{- 2} \ln (x)$ .

$\frac{1}{y^{- 2}} \frac{d y}{d x} = \frac{1}{y^{- 2}} (y^{- 2} (\ln (x)))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{y^{- 2}} \frac{d y}{d x} = \frac{1}{y^{- 2}} (y^{- 2} \ln (x))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{y^{- 2}} \frac{d y}{d x} = \ln (x)$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{- 2}} d y = \ln (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\int y^{2} d y = \int \ln (x) d x$

Step 2.2.2

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$\frac{1}{3} y^{3} + C_{1} = \int \ln (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

$\frac{1}{3} y^{3} + C_{1} = \ln (x) x - \int x \frac{1}{x} d x$

Step 2.3.2

Simplify.

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Step 2.3.2.1

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

$\frac{1}{3} y^{3} + C_{1} = \ln (x) x - \int \frac{x}{x} d x$

Step 2.3.2.2

Cancel the common factor of $x$ .

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Step 2.3.2.2.1

Cancel the common factor.

$\frac{1}{3} y^{3} + C_{1} = \ln (x) x - \int \frac{x}{x} d x$

Step 2.3.2.2.2

Rewrite the expression.

$\frac{1}{3} y^{3} + C_{1} = \ln (x) x - \int d x$

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

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

Step 2.3.5

Reorder terms.

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

Step 2.4

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

$\frac{1}{3} y^{3} = x \ln (x) - x + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (x \ln (x) - x + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $3 (\frac{1}{3} y^{3})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{3}$ and $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (x \ln (x) - x + K)$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$3 \frac{y^{3}}{3} = 3 (x \ln (x) - x + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (x \ln (x) - x + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (x \ln (x) - x + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{3} = 3 (x \ln (x)) + 3 (- x) + 3 K$

Step 3.2.2.1.2

Multiply $- 1$ by $3$ .

$y^{3} = 3 x \ln (x) - 3 x + 3 K$

Step 3.3

Simplify $3 \ln (x)$ by moving $3$ inside the logarithm.

$y^{3} = x \ln (x^{3}) - 3 x + 3 K$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{x \ln (x^{3}) - 3 x + 3 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{x \ln (x^{3}) - 3 x + K}$