Calculus Examples

$\frac{d y}{d x} + \frac{3}{x} y = \frac{\ln (x)}{x^{2}}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{3}{x}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{3}{x} d x}$

Step 1.2

Integrate $\frac{3}{x}$ .

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Step 1.2.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{3 \int \frac{1}{x} d x}$

Step 1.2.2

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

$e^{3 (\ln (| x |) + C)}$

Step 1.2.3

Simplify.

$e^{3 \ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{3 \ln (x)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (x^{3})}$

Step 1.5

Exponentiation and log are inverse functions.

$x^{3}$

Step 2

Multiply each term by the integrating factor $x^{3}$ .

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Step 2.1

Multiply each term by $x^{3}$ .

$x^{3} \frac{d y}{d x} + x^{3} (\frac{3}{x} y) = x^{3} \frac{\ln (x)}{x^{2}}$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{3}{x}$ and $y$ .

$x^{3} \frac{d y}{d x} + x^{3} \frac{3 y}{x} = x^{3} \frac{\ln (x)}{x^{2}}$

Step 2.2.2

Cancel the common factor of $x$ .

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Step 2.2.2.1

Factor $x$ out of $x^{3}$ .

$x^{3} \frac{d y}{d x} + x \cdot x^{2} \frac{3 y}{x} = x^{3} \frac{\ln (x)}{x^{2}}$

Step 2.2.2.2

Cancel the common factor.

$x^{3} \frac{d y}{d x} + x \cdot x^{2} \frac{3 y}{x} = x^{3} \frac{\ln (x)}{x^{2}}$

Step 2.2.2.3

Rewrite the expression.

$x^{3} \frac{d y}{d x} + x^{2} (3 y) = x^{3} \frac{\ln (x)}{x^{2}}$

Step 2.2.3

Rewrite using the commutative property of multiplication.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = x^{3} \frac{\ln (x)}{x^{2}}$

Step 2.3

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

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Step 2.3.1

Factor $x^{2}$ out of $x^{3}$ .

$x^{3} \frac{d y}{d x} + 3 x^{2} y = x^{2} x \frac{\ln (x)}{x^{2}}$

Step 2.3.2

Cancel the common factor.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = x^{2} x \frac{\ln (x)}{x^{2}}$

Step 2.3.3

Rewrite the expression.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = x \ln (x)$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [x^{3} y] = x \ln (x)$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x^{3} y] d x = \int x \ln (x) d x$

Step 5

Integrate the left side.

$x^{3} y = \int x \ln (x) d x$

Step 6

Integrate the right side.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Combine $\ln (x)$ and $\frac{x^{2}}{2}$ .

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

Step 6.3

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

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

Step 6.4

Simplify.

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Step 6.4.1

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

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

Step 6.4.2

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

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Step 6.4.2.1

Factor $x$ out of $x^{2}$ .

$x^{3} y = \frac{\ln (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x} d x)$

Step 6.4.2.2

Cancel the common factors.

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Step 6.4.2.2.1

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

$x^{3} y = \frac{\ln (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x^{1}} d x)$

Step 6.4.2.2.2

Factor $x$ out of $x^{1}$ .

$x^{3} y = \frac{\ln (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x \cdot 1} d x)$

Step 6.4.2.2.3

Cancel the common factor.

$x^{3} y = \frac{\ln (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x \cdot 1} d x)$

Step 6.4.2.2.4

Rewrite the expression.

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

Step 6.4.2.2.5

Divide $x$ by $1$ .

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

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

Step 6.5

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

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

Step 6.6

Simplify the answer.

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Step 6.6.1

Rewrite $\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C)$ as $\frac{1}{2} \ln (x) x^{2} - \frac{1}{2} \cdot \frac{1}{2} x^{2} + C$ .

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

Step 6.6.2

Simplify.

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Step 6.6.2.1

Combine $\frac{1}{2}$ and $\ln (x)$ .

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

Step 6.6.2.2

Combine $\frac{\ln (x)}{2}$ and $x^{2}$ .

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

Step 6.6.2.3

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

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

Step 6.6.2.4

Multiply $2$ by $2$ .

$x^{3} y = \frac{\ln (x) x^{2}}{2} - \frac{1}{4} x^{2} + C$

Step 6.6.3

Combine $x^{2}$ and $\frac{1}{4}$ .

$x^{3} y = \frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4} + C$

Step 6.6.4

Reorder terms.

$x^{3} y = \frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2} + C$

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Combine $x^{2}$ and $\frac{1}{4}$ .

$x^{3} y = \frac{1}{2} \cdot (\ln (x) x^{2}) - \frac{x^{2}}{4} + C$

Step 7.1.2

Remove parentheses.

$x^{3} y = \frac{1}{2} \cdot \ln (x) x^{2} - \frac{x^{2}}{4} + C$

Step 7.2

Divide each term in $x^{3} y = \frac{1}{2} \cdot (\ln (x) x^{2}) - \frac{x^{2}}{4} + C$ by $x^{3}$ and simplify.

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Step 7.2.1

Divide each term in $x^{3} y = \frac{1}{2} \cdot (\ln (x) x^{2}) - \frac{x^{2}}{4} + C$ by $x^{3}$ .

$\frac{x^{3} y}{x^{3}} = \frac{\frac{1}{2} \cdot (\ln (x) x^{2})}{x^{3}} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $x^{3}$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{x^{3} y}{x^{3}} = \frac{\frac{1}{2} \cdot (\ln (x) x^{2})}{x^{3}} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{2} \cdot (\ln (x) x^{2})}{x^{3}} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Simplify each term.

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Step 7.2.3.1.1

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 7.2.3.1.1.1

Factor $x^{2}$ out of $\frac{1}{2} \cdot (\ln (x) x^{2})$ .

$y = \frac{x^{2} (\frac{1}{2} \cdot (\ln (x)))}{x^{3}} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.1.2

Cancel the common factors.

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Step 7.2.3.1.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

$y = \frac{x^{2} (\frac{1}{2} \cdot (\ln (x)))}{x^{2} x} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{2} (\frac{1}{2} \cdot (\ln (x)))}{x^{2} x} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{2} \cdot (\ln (x))}{x} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.2

Simplify $\frac{1}{2} \ln (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$y = \frac{\ln (x^{\frac{1}{2}})}{x} + \frac{- \frac{x^{2}}{4}}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{\ln (x^{\frac{1}{2}})}{x} - \frac{x^{2}}{4} \cdot \frac{1}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.4

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

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Step 7.2.3.1.4.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$y = \frac{\ln (x^{\frac{1}{2}})}{x} + \frac{- x^{2}}{4} \cdot \frac{1}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.4.2

Factor $x^{2}$ out of $- x^{2}$ .

$y = \frac{\ln (x^{\frac{1}{2}})}{x} + \frac{x^{2} \cdot - 1}{4} \cdot \frac{1}{x^{3}} + \frac{C}{x^{3}}$

Step 7.2.3.1.4.3

Factor $x^{2}$ out of $x^{3}$ .

$y = \frac{\ln (x^{\frac{1}{2}})}{x} + \frac{x^{2} \cdot - 1}{4} \cdot \frac{1}{x^{2} x} + \frac{C}{x^{3}}$

Step 7.2.3.1.4.4

Cancel the common factor.

$y = \frac{\ln (x^{\frac{1}{2}})}{x} + \frac{x^{2} \cdot - 1}{4} \cdot \frac{1}{x^{2} x} + \frac{C}{x^{3}}$

Step 7.2.3.1.4.5

Rewrite the expression.

$y = \frac{\ln (x^{\frac{1}{2}})}{x} + \frac{- 1}{4} \cdot \frac{1}{x} + \frac{C}{x^{3}}$

Step 7.2.3.1.5

Multiply $\frac{- 1}{4}$ by $\frac{1}{x}$ .

$y = \frac{\ln (x^{\frac{1}{2}})}{x} + \frac{- 1}{4 x} + \frac{C}{x^{3}}$

Step 7.2.3.1.6

Move the negative in front of the fraction.

$y = \frac{\ln (x^{\frac{1}{2}})}{x} - \frac{1}{4 x} + \frac{C}{x^{3}}$