Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.2

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

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $\ln (x)$ .

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

Step 2.3.2.1.2

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

$\frac{1}{x}$

Step 2.3.2.2

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

$e^{y} + C_{1} = 2 \int u d u$

Step 2.3.3

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

$e^{y} + C_{1} = 2 (\frac{1}{2} u^{2} + C_{2})$

Step 2.3.4

Simplify.

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Step 2.3.4.1

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

$e^{y} + C_{1} = 2 (\frac{1}{2}) u^{2} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

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

$e^{y} + C_{1} = \frac{2}{2} u^{2} + C_{2}$

Step 2.3.4.2.2

Cancel the common factor of $2$ .

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Step 2.3.4.2.2.1

Cancel the common factor.

$e^{y} + C_{1} = \frac{2}{2} u^{2} + C_{2}$

Step 2.3.4.2.2.2

Rewrite the expression.

$e^{y} + C_{1} = 1 u^{2} + C_{2}$

Step 2.3.4.2.3

Multiply $u^{2}$ by $1$ .

$e^{y} + C_{1} = u^{2} + C_{2}$

Step 2.3.5

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

$e^{y} + C_{1} = \ln^{2} (x) + C_{2}$

Step 2.4

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

$e^{y} = \ln^{2} (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y}) = \ln (\ln^{2} (x) + K)$

Step 3.2

Expand the left side.

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Step 3.2.1

Expand $\ln (e^{y})$ by moving $y$ outside the logarithm.

$y \ln (e) = \ln (\ln^{2} (x) + K)$

Step 3.2.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (\ln^{2} (x) + K)$

Step 3.2.3

Multiply $y$ by $1$ .

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