Calculus Examples

$\frac{d y}{d x} = \frac{7 x y}{\ln^{8} (y)}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = 7 x \frac{y}{\ln^{8} (y)}$

Step 1.2

Multiply both sides by $\frac{\ln^{8} (y)}{y}$ .

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = \frac{\ln^{8} (y)}{y} (7 x \frac{y}{\ln^{8} (y)})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = 7 \frac{\ln^{8} (y)}{y} (x \frac{y}{\ln^{8} (y)})$

Step 1.3.2

Combine $7$ and $\frac{\ln^{8} (y)}{y}$ .

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = \frac{7 \ln^{8} (y)}{y} (x \frac{y}{\ln^{8} (y)})$

Step 1.3.3

Combine $x$ and $\frac{y}{\ln^{8} (y)}$ .

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = \frac{7 \ln^{8} (y)}{y} \cdot \frac{x y}{\ln^{8} (y)}$

Step 1.3.4

Combine.

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = \frac{7 \ln^{8} (y) (x y)}{y \ln^{8} (y)}$

Step 1.3.5

Cancel the common factor of $\ln^{8} (y)$ .

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Step 1.3.5.1

Cancel the common factor.

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = \frac{7 \ln^{8} (y) (x y)}{y \ln^{8} (y)}$

Step 1.3.5.2

Rewrite the expression.

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = \frac{7 (x y)}{y}$

Step 1.3.6

Cancel the common factor of $y$ .

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Step 1.3.6.1

Cancel the common factor.

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = \frac{7 (x y)}{y}$

Step 1.3.6.2

Divide $7 (x)$ by $1$ .

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = 7 (x)$

$\frac{\ln^{8} (y)}{y} \frac{d y}{d x} = 7 x$

Step 1.4

Rewrite the equation.

$\frac{\ln^{8} (y)}{y} d y = 7 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{\ln^{8} (y)}{y} d y = \int 7 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = \ln (y)$ . Then $d u = \frac{1}{y} d y$ , so $y d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = \ln (y)$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $\ln (y)$ .

$\frac{d}{d y} [\ln (y)]$

Step 2.2.1.1.2

The derivative of $\ln (y)$ with respect to $y$ is $\frac{1}{y}$ .

$\frac{1}{y}$

Step 2.2.1.2

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

$\int u^{8} d u = \int 7 x d x$

Step 2.2.2

By the Power Rule, the integral of $u^{8}$ with respect to $u$ is $\frac{1}{9} u^{9}$ .

$\frac{1}{9} u^{9} + C_{1} = \int 7 x d x$

Step 2.2.3

Replace all occurrences of $u$ with $\ln (y)$ .

$\frac{1}{9} \ln^{9} (y) + C_{1} = \int 7 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$\frac{1}{9} \ln^{9} (y) + C_{1} = 7 \int x d x$

Step 2.3.2

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

$\frac{1}{9} \ln^{9} (y) + C_{1} = 7 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $7 (\frac{1}{2} x^{2} + C_{2})$ as $7 (\frac{1}{2}) x^{2} + C_{2}$ .

$\frac{1}{9} \ln^{9} (y) + C_{1} = 7 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Combine $7$ and $\frac{1}{2}$ .

$\frac{1}{9} \ln^{9} (y) + C_{1} = \frac{7}{2} x^{2} + C_{2}$

Step 2.4

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

$\frac{1}{9} \ln^{9} (y) = \frac{7}{2} x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $9$ .

$9 (\frac{1}{9} \ln^{9} (y)) = 9 (\frac{7}{2} x^{2} + K)$

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 $9 (\frac{1}{9} \ln^{9} (y))$ .

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Step 3.2.1.1.1

Combine $\frac{1}{9}$ and $\ln^{9} (y)$ .

$9 \frac{\ln^{9} (y)}{9} = 9 (\frac{7}{2} x^{2} + K)$

Step 3.2.1.1.2

Cancel the common factor of $9$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$9 \frac{\ln^{9} (y)}{9} = 9 (\frac{7}{2} x^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$\ln^{9} (y) = 9 (\frac{7}{2} x^{2} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $9 (\frac{7}{2} x^{2} + K)$ .

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Step 3.2.2.1.1

Combine $\frac{7}{2}$ and $x^{2}$ .

$\ln^{9} (y) = 9 (\frac{7 x^{2}}{2} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$\ln^{9} (y) = 9 \frac{7 x^{2}}{2} + 9 K$

Step 3.2.2.1.3

Multiply $9 \frac{7 x^{2}}{2}$ .

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Step 3.2.2.1.3.1

Combine $9$ and $\frac{7 x^{2}}{2}$ .

$\ln^{9} (y) = \frac{9 (7 x^{2})}{2} + 9 K$

Step 3.2.2.1.3.2

Multiply $7$ by $9$ .

$\ln^{9} (y) = \frac{63 x^{2}}{2} + 9 K$

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

$\ln (y) = \sqrt[9]{\frac{63 x^{2}}{2} + 9 K}$

Step 3.3.2

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (y)} = e^{\sqrt[9]{\frac{63 x^{2}}{2} + 9 K}}$

Step 3.3.3

Rewrite $\ln (y) = \sqrt[9]{\frac{63 x^{2}}{2} + 9 K}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\sqrt[9]{\frac{63 x^{2}}{2} + 9 K}} = y$

Step 3.3.4

Solve for $y$ .

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Step 3.3.4.1

Rewrite the equation as $y = e^{\sqrt[9]{\frac{63 x^{2}}{2} + 9 K}}$ .

$y = e^{\sqrt[9]{\frac{63 x^{2}}{2} + 9 K}}$

Step 3.3.4.2

Simplify $e^{\sqrt[9]{\frac{63 x^{2}}{2} + 9 K}}$ .

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Step 3.3.4.2.1

Factor $9$ out of $\frac{63 x^{2}}{2} + 9 K$ .

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Step 3.3.4.2.1.1

Factor $9$ out of $\frac{63 x^{2}}{2}$ .

$y = e^{\sqrt[9]{9 \frac{7 x^{2}}{2} + 9 K}}$

Step 3.3.4.2.1.2

Factor $9$ out of $9 K$ .

$y = e^{\sqrt[9]{9 \frac{7 x^{2}}{2} + 9 K}}$

Step 3.3.4.2.1.3

Factor $9$ out of $9 \frac{7 x^{2}}{2} + 9 K$ .

$y = e^{\sqrt[9]{9 (\frac{7 x^{2}}{2} + K)}}$

Step 3.3.4.2.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = e^{\sqrt[9]{9 (\frac{7 x^{2}}{2} + K \cdot \frac{2}{2})}}$

Step 3.3.4.2.3

Simplify terms.

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Step 3.3.4.2.3.1

Combine $K$ and $\frac{2}{2}$ .

$y = e^{\sqrt[9]{9 (\frac{7 x^{2}}{2} + \frac{K \cdot 2}{2})}}$

Step 3.3.4.2.3.2

Combine the numerators over the common denominator.

$y = e^{\sqrt[9]{9 \frac{7 x^{2} + K \cdot 2}{2}}}$

Step 3.3.4.2.4

Move $2$ to the left of $K$ .

$y = e^{\sqrt[9]{9 \frac{7 x^{2} + 2 K}{2}}}$

Step 3.3.4.2.5

Combine $9$ and $\frac{7 x^{2} + 2 K}{2}$ .

$y = e^{\sqrt[9]{\frac{9 (7 x^{2} + 2 K)}{2}}}$

Step 3.3.4.2.6

Rewrite $\sqrt[9]{\frac{9 (7 x^{2} + 2 K)}{2}}$ as $\frac{\sqrt[9]{9 (7 x^{2} + 2 K)}}{\sqrt[9]{2}}$ .

$y = e^{\frac{\sqrt[9]{9 (7 x^{2} + 2 K)}}{\sqrt[9]{2}}}$

Step 4

Simplify the constant of integration.

$y = e^{\frac{\sqrt[9]{9 (7 x^{2} + K)}}{\sqrt[9]{2}}}$