Calculus Examples

$x \cdot \frac{d y}{d x} + y = x^{4} \ln (x)$

Step 1

Check if the left side of the equation is the result of the derivative of the term $x y$ .

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$x \frac{d}{d x} [y] + y \frac{d}{d x} [x]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x y' + y \frac{d}{d x} [x]$

Step 1.3

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

$x y' + y \cdot 1$

Step 1.4

Substitute $\frac{d y}{d x}$ for $y'$ .

$x \frac{d y}{d x} + y \cdot 1$

Step 1.5

Reorder $y$ and $1$ .

$x \frac{d y}{d x} + 1 \cdot y$

Step 1.6

Multiply $y$ by $1$ .

$x \frac{d y}{d x} + y$

Step 2

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

$\frac{d}{d x} [x y] = x^{4} \ln (x)$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [x y] d x = \int x^{4} \ln (x) d x$

Step 4

Integrate the left side.

$x y = \int x^{4} \ln (x) d x$

Step 5

Integrate the right side.

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Step 5.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{4}$ .

$x y = \ln (x) (\frac{1}{5} x^{5}) - \int \frac{1}{5} x^{5} \frac{1}{x} d x$

Step 5.2

Simplify.

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Step 5.2.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$x y = \ln (x) \frac{x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x$

Step 5.2.2

Combine $\ln (x)$ and $\frac{x^{5}}{5}$ .

$x y = \frac{\ln (x) x^{5}}{5} - \int \frac{1}{5} x^{5} \frac{1}{x} d x$

Step 5.3

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int x^{5} \frac{1}{x} d x)$

Step 5.4

Simplify.

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Step 5.4.1

Combine $x^{5}$ and $\frac{1}{x}$ .

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x^{5}}{x} d x)$

Step 5.4.2

Cancel the common factor of $x^{5}$ and $x$ .

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Step 5.4.2.1

Factor $x$ out of $x^{5}$ .

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x} d x)$

Step 5.4.2.2

Cancel the common factors.

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Step 5.4.2.2.1

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

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x^{1}} d x)$

Step 5.4.2.2.2

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

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x \cdot 1} d x)$

Step 5.4.2.2.3

Cancel the common factor.

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x \cdot x^{4}}{x \cdot 1} d x)$

Step 5.4.2.2.4

Rewrite the expression.

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int \frac{x^{4}}{1} d x)$

Step 5.4.2.2.5

Divide $x^{4}$ by $1$ .

$x y = \frac{\ln (x) x^{5}}{5} - (\frac{1}{5} \int x^{4} d x)$

$x y = \frac{\ln (x) x^{5}}{5} - \frac{1}{5} \int x^{4} d x$

Step 5.5

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$x y = \frac{\ln (x) x^{5}}{5} - \frac{1}{5} (\frac{1}{5} x^{5} + C)$

Step 5.6

Simplify the answer.

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Step 5.6.1

Rewrite $\frac{\ln (x) x^{5}}{5} - \frac{1}{5} (\frac{1}{5} x^{5} + C)$ as $\frac{1}{5} \ln (x) x^{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$ .

$x y = \frac{1}{5} \ln (x) x^{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$

Step 5.6.2

Simplify.

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Step 5.6.2.1

Combine $\frac{1}{5}$ and $\ln (x)$ .

$x y = \frac{\ln (x)}{5} x^{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$

Step 5.6.2.2

Combine $\frac{\ln (x)}{5}$ and $x^{5}$ .

$x y = \frac{\ln (x) x^{5}}{5} - \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$

Step 5.6.2.3

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$x y = \frac{\ln (x) x^{5}}{5} - \frac{1}{5 \cdot 5} x^{5} + C$

Step 5.6.2.4

Multiply $5$ by $5$ .

$x y = \frac{\ln (x) x^{5}}{5} - \frac{1}{25} x^{5} + C$

$x y = \frac{1}{5} \ln (x) x^{5} - \frac{1}{25} x^{5} + C$

Step 6

Solve for $y$ .

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Step 6.1

Simplify.

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Step 6.1.1

Combine $x^{5}$ and $\frac{1}{25}$ .

$x y = \frac{1}{5} \cdot (\ln (x) x^{5}) - \frac{x^{5}}{25} + C$

Step 6.1.2

Remove parentheses.

$x y = \frac{1}{5} \cdot \ln (x) x^{5} - \frac{x^{5}}{25} + C$

Step 6.2

Divide each term in $x y = \frac{1}{5} \cdot (\ln (x) x^{5}) - \frac{x^{5}}{25} + C$ by $x$ and simplify.

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Step 6.2.1

Divide each term in $x y = \frac{1}{5} \cdot (\ln (x) x^{5}) - \frac{x^{5}}{25} + C$ by $x$ .

$\frac{x y}{x} = \frac{\frac{1}{5} \cdot (\ln (x) x^{5})}{x} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $x$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{\frac{1}{5} \cdot (\ln (x) x^{5})}{x} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{5} \cdot (\ln (x) x^{5})}{x} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Cancel the common factor of $x^{5}$ and $x$ .

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Step 6.2.3.1.1.1

Factor $x$ out of $\frac{1}{5} \cdot (\ln (x) x^{5})$ .

$y = \frac{x (\frac{1}{5} \cdot (\ln (x) x^{4}))}{x} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.1.2

Cancel the common factors.

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Step 6.2.3.1.1.2.1

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

$y = \frac{x (\frac{1}{5} \cdot (\ln (x) x^{4}))}{x^{1}} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.1.2.2

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

$y = \frac{x (\frac{1}{5} \cdot (\ln (x) x^{4}))}{x \cdot 1} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.1.2.3

Cancel the common factor.

$y = \frac{x (\frac{1}{5} \cdot (\ln (x) x^{4}))}{x \cdot 1} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.1.2.4

Rewrite the expression.

$y = \frac{\frac{1}{5} \cdot (\ln (x) x^{4})}{1} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.1.2.5

Divide $\frac{1}{5} \cdot (\ln (x) x^{4})$ by $1$ .

$y = \frac{1}{5} \cdot (\ln (x) x^{4}) + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.2

Multiply $\frac{1}{5} (\ln (x) x^{4})$ .

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Step 6.2.3.1.2.1

Reorder $\ln (x)$ and $\frac{1}{5}$ .

$y = \frac{1}{5} \ln (x) x^{4} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.2.2

Simplify $\frac{1}{5} \ln (x)$ by moving $\frac{1}{5}$ inside the logarithm.

$y = \ln (x^{\frac{1}{5}}) x^{4} + \frac{- \frac{x^{5}}{25}}{x} + \frac{C}{x}$

Step 6.2.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \ln (x^{\frac{1}{5}}) x^{4} - \frac{x^{5}}{25} \cdot \frac{1}{x} + \frac{C}{x}$

Step 6.2.3.1.4

Cancel the common factor of $x$ .

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Step 6.2.3.1.4.1

Move the leading negative in $- \frac{x^{5}}{25}$ into the numerator.

$y = \ln (x^{\frac{1}{5}}) x^{4} + \frac{- x^{5}}{25} \cdot \frac{1}{x} + \frac{C}{x}$

Step 6.2.3.1.4.2

Factor $x$ out of $- x^{5}$ .

$y = \ln (x^{\frac{1}{5}}) x^{4} + \frac{x (- x^{4})}{25} \cdot \frac{1}{x} + \frac{C}{x}$

Step 6.2.3.1.4.3

Cancel the common factor.

$y = \ln (x^{\frac{1}{5}}) x^{4} + \frac{x (- x^{4})}{25} \cdot \frac{1}{x} + \frac{C}{x}$

Step 6.2.3.1.4.4

Rewrite the expression.

$y = \ln (x^{\frac{1}{5}}) x^{4} + \frac{- x^{4}}{25} + \frac{C}{x}$

Step 6.2.3.1.5

Move the negative in front of the fraction.

$y = \ln (x^{\frac{1}{5}}) x^{4} - \frac{x^{4}}{25} + \frac{C}{x}$

Step 6.2.3.2

Reorder factors in $\ln (x^{\frac{1}{5}}) x^{4} - \frac{x^{4}}{25} + \frac{C}{x}$ .

$y = x^{4} \ln (x^{\frac{1}{5}}) - \frac{x^{4}}{25} + \frac{C}{x}$