Calculus Examples

$\frac{d y}{d x} = \frac{8 e^{4 x}}{2 \cos (2 y)}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\cos (2 y)$ .

$\cos (2 y) \frac{d y}{d x} = \cos (2 y) \frac{8 e^{4 x}}{2 \cos (2 y)}$

Step 1.2

Simplify.

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Step 1.2.1

Cancel the common factor of $\cos (2 y)$ .

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Step 1.2.1.1

Factor $\cos (2 y)$ out of $2 \cos (2 y)$ .

$\cos (2 y) \frac{d y}{d x} = \cos (2 y) \frac{8 e^{4 x}}{\cos (2 y) \cdot 2}$

Step 1.2.1.2

Cancel the common factor.

$\cos (2 y) \frac{d y}{d x} = \cos (2 y) \frac{8 e^{4 x}}{\cos (2 y) \cdot 2}$

Step 1.2.1.3

Rewrite the expression.

$\cos (2 y) \frac{d y}{d x} = \frac{8 e^{4 x}}{2}$

Step 1.2.2

Cancel the common factor of $8$ and $2$ .

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Step 1.2.2.1

Factor $2$ out of $8 e^{4 x}$ .

$\cos (2 y) \frac{d y}{d x} = \frac{2 (4 e^{4 x})}{2}$

Step 1.2.2.2

Cancel the common factors.

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Step 1.2.2.2.1

Factor $2$ out of $2$ .

$\cos (2 y) \frac{d y}{d x} = \frac{2 (4 e^{4 x})}{2 (1)}$

Step 1.2.2.2.2

Cancel the common factor.

$\cos (2 y) \frac{d y}{d x} = \frac{2 (4 e^{4 x})}{2 \cdot 1}$

Step 1.2.2.2.3

Rewrite the expression.

$\cos (2 y) \frac{d y}{d x} = \frac{4 e^{4 x}}{1}$

Step 1.2.2.2.4

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

$\cos (2 y) \frac{d y}{d x} = 4 e^{4 x}$

Step 1.3

Rewrite the equation.

$\cos (2 y) d y = 4 e^{4 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \cos (2 y) d y = \int 4 e^{4 x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Let $u_{1} = 2 y$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.1.1.1

Differentiate $2 y$ .

$\frac{d}{d y} [2 y]$

Step 2.2.1.1.2

Since $2$ is constant with respect to $y$ , the derivative of $2 y$ with respect to $y$ is $2 \frac{d}{d y} [y]$ .

$2 \frac{d}{d y} [y]$

Step 2.2.1.1.3

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

$2 \cdot 1$

Step 2.2.1.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2.1.2

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

$\int \cos (u_{1}) \frac{1}{2} d u_{1} = \int 4 e^{4 x} d x$

Step 2.2.2

Combine $\cos (u_{1})$ and $\frac{1}{2}$ .

$\int \frac{\cos (u_{1})}{2} d u_{1} = \int 4 e^{4 x} d x$

Step 2.2.3

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

$\frac{1}{2} \int \cos (u_{1}) d u_{1} = \int 4 e^{4 x} d x$

Step 2.2.4

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

$\frac{1}{2} (\sin (u_{1}) + C_{1}) = \int 4 e^{4 x} d x$

Step 2.2.5

Simplify.

$\frac{1}{2} \sin (u_{1}) + C_{1} = \int 4 e^{4 x} d x$

Step 2.2.6

Replace all occurrences of $u_{1}$ with $2 y$ .

$\frac{1}{2} \sin (2 y) + C_{1} = \int 4 e^{4 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{2} \sin (2 y) + C_{1} = 4 \int e^{4 x} d x$

Step 2.3.2

Let $u_{2} = 4 x$ . Then $d u_{2} = 4 d x$ , so $\frac{1}{4} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.2.1

Let $u_{2} = 4 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 2.3.2.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 2.3.2.1.3

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

$4 \cdot 1$

Step 2.3.2.1.4

Multiply $4$ by $1$ .

$4$

Step 2.3.2.2

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

$\frac{1}{2} \sin (2 y) + C_{1} = 4 \int e^{u_{2}} \frac{1}{4} d u_{2}$

Step 2.3.3

Combine $e^{u_{2}}$ and $\frac{1}{4}$ .

$\frac{1}{2} \sin (2 y) + C_{1} = 4 \int \frac{e^{u_{2}}}{4} d u_{2}$

Step 2.3.4

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{2} \sin (2 y) + C_{1} = 4 (\frac{1}{4} \int e^{u_{2}} d u_{2})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{4}$ and $4$ .

$\frac{1}{2} \sin (2 y) + C_{1} = \frac{4}{4} \int e^{u_{2}} d u_{2}$

Step 2.3.5.2

Cancel the common factor of $4$ .

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Step 2.3.5.2.1

Cancel the common factor.

$\frac{1}{2} \sin (2 y) + C_{1} = \frac{4}{4} \int e^{u_{2}} d u_{2}$

Step 2.3.5.2.2

Rewrite the expression.

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

Step 2.3.5.3

Multiply $\int e^{u_{2}} d u_{2}$ by $1$ .

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

Step 2.3.6

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

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

Step 2.3.7

Replace all occurrences of $u_{2}$ with $4 x$ .

$\frac{1}{2} \sin (2 y) + C_{1} = e^{4 x} + C_{2}$

Step 2.4

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

$\frac{1}{2} \sin (2 y) = e^{4 x} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \sin (2 y)) = 2 (e^{4 x} + K)$

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 $2 (\frac{1}{2} \sin (2 y))$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $\sin (2 y)$ .

$2 \frac{\sin (2 y)}{2} = 2 (e^{4 x} + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{\sin (2 y)}{2} = 2 (e^{4 x} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$\sin (2 y) = 2 (e^{4 x} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$\sin (2 y) = 2 e^{4 x} + 2 K$

Step 3.3

Reorder $2 e^{4 x}$ and $2 K$ .

$\sin (2 y) = 2 K + 2 e^{4 x}$

Step 3.4

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$2 y = \arcsin (2 K + 2 e^{4 x})$

Step 3.5

Divide each term in $2 y = \arcsin (2 K + 2 e^{4 x})$ by $2$ and simplify.

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Step 3.5.1

Divide each term in $2 y = \arcsin (2 K + 2 e^{4 x})$ by $2$ .

$\frac{2 y}{2} = \frac{\arcsin (2 K + 2 e^{4 x})}{2}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\arcsin (2 K + 2 e^{4 x})}{2}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (2 K + 2 e^{4 x})}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\arcsin (K + 2 e^{4 x})}{2}$