Calculus Examples

$e^{5 y} d y = e^{5 x} d x$

Step 1

Integrate both sides.

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Step 1.1

Set up an integral on each side.

$\int e^{5 y} d y = \int e^{5 x} d x$

Step 1.2

Integrate the left side.

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Step 1.2.1

Let $u_{1} = 5 y$ . Then $d u_{1} = 5 d y$ , so $\frac{1}{5} d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.2.1.1

Let $u_{1} = 5 y$ . Find $\frac{d u_{1}}{d y}$ .

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Step 1.2.1.1.1

Differentiate $5 y$ .

$\frac{d}{d y} [5 y]$

Step 1.2.1.1.2

Since $5$ is constant with respect to $y$ , the derivative of $5 y$ with respect to $y$ is $5 \frac{d}{d y} [y]$ .

$5 \frac{d}{d y} [y]$

Step 1.2.1.1.3

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

$5 \cdot 1$

Step 1.2.1.1.4

Multiply $5$ by $1$ .

$5$

Step 1.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int e^{u_{1}} \frac{1}{5} d u_{1} = \int e^{5 x} d x$

Step 1.2.2

Combine $e^{u_{1}}$ and $\frac{1}{5}$ .

$\int \frac{e^{u_{1}}}{5} d u_{1} = \int e^{5 x} d x$

Step 1.2.3

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

$\frac{1}{5} \int e^{u_{1}} d u_{1} = \int e^{5 x} d x$

Step 1.2.4

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$\frac{1}{5} (e^{u_{1}} + C_{1}) = \int e^{5 x} d x$

Step 1.2.5

Simplify.

$\frac{1}{5} e^{u_{1}} + C_{1} = \int e^{5 x} d x$

Step 1.2.6

Replace all occurrences of $u_{1}$ with $5 y$ .

$\frac{1}{5} e^{5 y} + C_{1} = \int e^{5 x} d x$

Step 1.3

Integrate the right side.

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Step 1.3.1

Let $u_{2} = 5 x$ . Then $d u_{2} = 5 d x$ , so $\frac{1}{5} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.3.1.1

Let $u_{2} = 5 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.3.1.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 1.3.1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 1.3.1.1.3

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

$5 \cdot 1$

Step 1.3.1.1.4

Multiply $5$ by $1$ .

$5$

Step 1.3.1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{5} e^{5 y} + C_{1} = \int e^{u_{2}} \frac{1}{5} d u_{2}$

Step 1.3.2

Combine $e^{u_{2}}$ and $\frac{1}{5}$ .

$\frac{1}{5} e^{5 y} + C_{1} = \int \frac{e^{u_{2}}}{5} d u_{2}$

Step 1.3.3

Since $\frac{1}{5}$ is constant with respect to $u_{2}$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} e^{5 y} + C_{1} = \frac{1}{5} \int e^{u_{2}} d u_{2}$

Step 1.3.4

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{5} e^{5 y} + C_{1} = \frac{1}{5} (e^{u_{2}} + C_{2})$

Step 1.3.5

Simplify.

$\frac{1}{5} e^{5 y} + C_{1} = \frac{1}{5} e^{u_{2}} + C_{2}$

Step 1.3.6

Replace all occurrences of $u_{2}$ with $5 x$ .

$\frac{1}{5} e^{5 y} + C_{1} = \frac{1}{5} e^{5 x} + C_{2}$

Step 1.4

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

$\frac{1}{5} e^{5 y} = \frac{1}{5} e^{5 x} + K$

Step 2

Solve for $y$ .

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Step 2.1

Multiply both sides of the equation by $5$ .

$5 (\frac{1}{5} e^{5 y}) = 5 (\frac{1}{5} e^{5 x} + K)$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $5 (\frac{1}{5} e^{5 y})$ .

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Step 2.2.1.1.1

Combine $\frac{1}{5}$ and $e^{5 y}$ .

$5 \frac{e^{5 y}}{5} = 5 (\frac{1}{5} e^{5 x} + K)$

Step 2.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$5 \frac{e^{5 y}}{5} = 5 (\frac{1}{5} e^{5 x} + K)$

Step 2.2.1.1.2.2

Rewrite the expression.

$e^{5 y} = 5 (\frac{1}{5} e^{5 x} + K)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $5 (\frac{1}{5} e^{5 x} + K)$ .

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Step 2.2.2.1.1

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

$e^{5 y} = 5 (\frac{e^{5 x}}{5} + K)$

Step 2.2.2.1.2

Apply the distributive property.

$e^{5 y} = 5 \frac{e^{5 x}}{5} + 5 K$

Step 2.2.2.1.3

Cancel the common factor of $5$ .

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Step 2.2.2.1.3.1

Cancel the common factor.

$e^{5 y} = 5 \frac{e^{5 x}}{5} + 5 K$

Step 2.2.2.1.3.2

Rewrite the expression.

$e^{5 y} = e^{5 x} + 5 K$

Step 2.3

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

$\ln (e^{5 y}) = \ln (e^{5 x} + 5 K)$

Step 2.4

Expand the left side.

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Step 2.4.1

Expand $\ln (e^{5 y})$ by moving $5 y$ outside the logarithm.

$5 y \ln (e) = \ln (e^{5 x} + 5 K)$

Step 2.4.2

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

$5 y \cdot 1 = \ln (e^{5 x} + 5 K)$

Step 2.4.3

Multiply $5$ by $1$ .

$5 y = \ln (e^{5 x} + 5 K)$

Step 2.5

Divide each term in $5 y = \ln (e^{5 x} + 5 K)$ by $5$ and simplify.

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Step 2.5.1

Divide each term in $5 y = \ln (e^{5 x} + 5 K)$ by $5$ .

$\frac{5 y}{5} = \frac{\ln (e^{5 x} + 5 K)}{5}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{\ln (e^{5 x} + 5 K)}{5}$

Step 2.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (e^{5 x} + 5 K)}{5}$

Step 3

Simplify the constant of integration.

$y = \frac{\ln (e^{5 x} + K)}{5}$