Calculus Examples

$\frac{d y}{d x} + 6 y + \ln (x) = 0$

Step 1

Subtract $\ln (x)$ from both sides of the equation.

$\frac{d y}{d x} + 6 y = - \ln (x)$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 6$ .

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Step 2.1

Set up the integration.

$e^{\int 6 d x}$

Step 2.2

Apply the constant rule.

$e^{6 x + C}$

Step 2.3

Remove the constant of integration.

$e^{6 x}$

Step 3

Multiply each term by the integrating factor $e^{6 x}$ .

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Step 3.1

Multiply each term by $e^{6 x}$ .

$e^{6 x} \frac{d y}{d x} + e^{6 x} (6 y) = e^{6 x} (- \ln (x))$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{6 x} \frac{d y}{d x} + 6 e^{6 x} y = e^{6 x} (- \ln (x))$

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{6 x} \frac{d y}{d x} + 6 e^{6 x} y = - e^{6 x} \ln (x)$

Step 3.4

Reorder factors in $e^{6 x} \frac{d y}{d x} + 6 e^{6 x} y = - e^{6 x} \ln (x)$ .

$e^{6 x} \frac{d y}{d x} + 6 y e^{6 x} = - e^{6 x} \ln (x)$

Step 4

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

$\frac{d}{d x} [e^{6 x} y] = - e^{6 x} \ln (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{6 x} y] d x = \int - e^{6 x} \ln (x) d x$

Step 6

Integrate the left side.

$e^{6 x} y = \int - e^{6 x} \ln (x) d x$

Step 7

Integrate the right side.

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Step 7.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{6 x} y = - \int e^{6 x} \ln (x) d x$

Step 7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = e^{6 x}$ .

$e^{6 x} y = - (\ln (x) (\frac{1}{6} e^{6 x}) - \int \frac{1}{6} e^{6 x} \frac{1}{x} d x)$

Step 7.3

Simplify.

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Step 7.3.1

Combine $\frac{1}{6}$ and $e^{6 x}$ .

$e^{6 x} y = - (\ln (x) \frac{e^{6 x}}{6} - \int \frac{1}{6} e^{6 x} \frac{1}{x} d x)$

Step 7.3.2

Combine $\ln (x)$ and $\frac{e^{6 x}}{6}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \int \frac{1}{6} e^{6 x} \frac{1}{x} d x)$

Step 7.3.3

Combine $\frac{1}{6}$ and $e^{6 x}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \int \frac{e^{6 x}}{6} \cdot \frac{1}{x} d x)$

Step 7.3.4

Multiply $\frac{e^{6 x}}{6}$ by $\frac{1}{x}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \int \frac{e^{6 x}}{6 x} d x)$

Step 7.4

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - (\frac{1}{6} \int \frac{e^{6 x}}{x} d x))$

Step 7.5

Let $u = 6 x$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.5.1

Let $u = 6 x$ . Find $\frac{d u}{d x}$ .

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Step 7.5.1.1

Differentiate $6 x$ .

$\frac{d}{d x} [6 x]$

Step 7.5.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x]$

Step 7.5.1.3

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

$6 \cdot 1$

Step 7.5.1.4

Multiply $6$ by $1$ .

$6$

Step 7.5.2

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

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int \frac{e^{u}}{\frac{u}{6}} \cdot \frac{1}{6} d u)$

Step 7.6

Simplify.

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Step 7.6.1

Multiply by the reciprocal of the fraction to divide by $\frac{u}{6}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int e^{u} \frac{6}{u} \frac{1}{6} d u)$

Step 7.6.2

Combine $e^{u}$ and $\frac{6}{u}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int \frac{e^{u} \cdot 6}{u} \cdot \frac{1}{6} d u)$

Step 7.6.3

Move $6$ to the left of $e^{u}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int \frac{6 \cdot e^{u}}{u} \cdot \frac{1}{6} d u)$

Step 7.6.4

Multiply $\frac{6 e^{u}}{u}$ by $\frac{1}{6}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int \frac{6 e^{u}}{u \cdot 6} d u)$

Step 7.6.5

Move $6$ to the left of $u$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int \frac{6 e^{u}}{6 \cdot u} d u)$

Step 7.6.6

Cancel the common factor of $6$ .

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Step 7.6.6.1

Cancel the common factor.

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int \frac{6 e^{u}}{6 u} d u)$

Step 7.6.6.2

Rewrite the expression.

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} \int \frac{e^{u}}{u} d u)$

Step 7.7

$\int \frac{e^{u}}{u} d u$ is a special integral. The integral is the exponential integral function.

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} (E i (u) + C))$

Step 7.8

Rewrite $- (\frac{\ln (x) e^{6 x}}{6} - \frac{1}{6} (E i (u) + C))$ as $- (\frac{\ln (x) e^{6 x}}{6} - \frac{E i (u)}{6}) + C$ .

$e^{6 x} y = - (\frac{1}{6} \ln (x) e^{6 x} - \frac{E i (u)}{6}) + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

Multiply $\frac{1}{6} (\ln (x) e^{6 x})$ .

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Step 8.1.1.1

Combine $\ln (x)$ and $\frac{1}{6}$ .

$e^{6 x} y = - (\frac{\ln (x)}{6} e^{6 x} - \frac{E i (u)}{6}) + C$

Step 8.1.1.2

Combine $\frac{\ln (x)}{6}$ and $e^{6 x}$ .

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{E i (u)}{6}) + C$

$e^{6 x} y = - (\frac{\ln (x) e^{6 x}}{6} - \frac{E i u}{6}) + C$

Step 8.1.2

Apply the distributive property.

$e^{6 x} y = - \frac{\ln (x) e^{6 x}}{6} - - \frac{E i u}{6} + C$

Step 8.1.3

Multiply $- - \frac{E i u}{6}$ .

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Step 8.1.3.1

Multiply $- 1$ by $- 1$ .

$e^{6 x} y = - \frac{\ln (x) e^{6 x}}{6} + 1 \frac{E i u}{6} + C$

Step 8.1.3.2

Multiply $\frac{E i u}{6}$ by $1$ .

$e^{6 x} y = - \frac{\ln (x) e^{6 x}}{6} + \frac{E i u}{6} + C$

Step 8.1.4

Reorder factors in $- \frac{\ln (x) e^{6 x}}{6} + \frac{E i u}{6}$ .

$e^{6 x} y = - \frac{e^{6 x} \ln (x)}{6} + \frac{E i u}{6} + C$

Step 8.2

Divide each term in $e^{6 x} y = - \frac{e^{6 x} \ln (x)}{6} + \frac{E i u}{6} + C$ by $e^{6 x}$ and simplify.

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Step 8.2.1

Divide each term in $e^{6 x} y = - \frac{e^{6 x} \ln (x)}{6} + \frac{E i u}{6} + C$ by $e^{6 x}$ .

$\frac{e^{6 x} y}{e^{6 x}} = \frac{- \frac{e^{6 x} \ln (x)}{6}}{e^{6 x}} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $e^{6 x}$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{e^{6 x} y}{e^{6 x}} = \frac{- \frac{e^{6 x} \ln (x)}{6}}{e^{6 x}} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{e^{6 x} \ln (x)}{6}}{e^{6 x}} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{e^{6 x} \ln (x)}{6} \cdot \frac{1}{e^{6 x}} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.2

Cancel the common factor of $e^{6 x}$ .

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Step 8.2.3.1.2.1

Move the leading negative in $- \frac{e^{6 x} \ln (x)}{6}$ into the numerator.

$y = \frac{- e^{6 x} \ln (x)}{6} \cdot \frac{1}{e^{6 x}} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.2.2

Factor $e^{6 x}$ out of $- e^{6 x} \ln (x)$ .

$y = \frac{e^{6 x} (- 1 \ln (x))}{6} \cdot \frac{1}{e^{6 x}} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.2.3

Cancel the common factor.

$y = \frac{e^{6 x} (- 1 \ln (x))}{6} \cdot \frac{1}{e^{6 x}} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.2.4

Rewrite the expression.

$y = \frac{- 1 \ln (x)}{6} + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.3

Rewrite $\frac{- 1 \ln (x)}{6}$ as $\frac{- 1}{6} \ln (x)$ .

$y = \frac{- 1}{6} \ln (x) + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.4

Simplify $\frac{- 1}{6} \ln (x)$ by moving $- \frac{- 1}{6}$ inside the logarithm.

$y = - \ln (x^{- \frac{- 1}{6}}) + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.5

Move the negative in front of the fraction.

$y = - \ln (x^{- - \frac{1}{6}}) + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.6

Multiply $- - \frac{1}{6}$ .

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Step 8.2.3.1.6.1

Multiply $- 1$ by $- 1$ .

$y = - \ln (x^{1 (\frac{1}{6})}) + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.6.2

Multiply $\frac{1}{6}$ by $1$ .

$y = - \ln (x^{\frac{1}{6}}) + \frac{\frac{E i u}{6}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.7

Multiply the numerator by the reciprocal of the denominator.

$y = - \ln (x^{\frac{1}{6}}) + \frac{E i u}{6} \cdot \frac{1}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.8

Combine.

$y = - \ln (x^{\frac{1}{6}}) + \frac{E i u \cdot 1}{6 e^{6 x}} + \frac{C}{e^{6 x}}$

Step 8.2.3.1.9

Multiply $E$ by $1$ .

$y = - \ln (x^{\frac{1}{6}}) + \frac{E i u}{6 e^{6 x}} + \frac{C}{e^{6 x}}$