Calculus Examples

$(e^{2} + x + 1) d x + (\sin (y) + 2 \cos (y)) d y = 0$

Step 1

Subtract $(e^{2} + x + 1) d x$ from both sides of the equation.

$(\sin (y) + 2 \cos (y)) d y = - (e^{2} + x + 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \sin (y) + 2 \cos (y) d y = \int - (e^{2} + x + 1) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int \sin (y) d y + \int 2 \cos (y) d y = \int - (e^{2} + x + 1) d x$

Step 2.2.2

The integral of $\sin (y)$ with respect to $y$ is $- \cos (y)$ .

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

Step 2.2.3

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

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

Step 2.2.4

The integral of $\cos (y)$ with respect to $y$ is $\sin (y)$ .

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

Step 2.2.5

Simplify.

$- \cos (y) + 2 \sin (y) + C_{3} = \int - (e^{2} + x + 1) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Multiply $- (e^{2} + x + 1)$ .

$- \cos (y) + 2 \sin (y) + C_{3} = \int - e^{2} - x - 1 \cdot 1 d x$

Step 2.3.2

Multiply $- 1$ by $1$ .

$- \cos (y) + 2 \sin (y) + C_{3} = \int - e^{2} - x - 1 d x$

Step 2.3.3

Split the single integral into multiple integrals.

$- \cos (y) + 2 \sin (y) + C_{3} = \int - e^{2} d x + \int - x d x + \int - 1 d x$

Step 2.3.4

Apply the constant rule.

$- \cos (y) + 2 \sin (y) + C_{3} = - e^{2} x + C_{4} + \int - x d x + \int - 1 d x$

Step 2.3.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \cos (y) + 2 \sin (y) + C_{3} = - e^{2} x + C_{4} - \int x d x + \int - 1 d x$

Step 2.3.6

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

$- \cos (y) + 2 \sin (y) + C_{3} = - e^{2} x + C_{4} - (\frac{1}{2} x^{2} + C_{5}) + \int - 1 d x$

Step 2.3.7

Apply the constant rule.

$- \cos (y) + 2 \sin (y) + C_{3} = - e^{2} x + C_{4} - (\frac{1}{2} x^{2} + C_{5}) - x + C_{6}$

Step 2.3.8

Simplify.

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Step 2.3.8.1

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

$- \cos (y) + 2 \sin (y) + C_{3} = - e^{2} x + C_{4} - (\frac{x^{2}}{2} + C_{5}) - x + C_{6}$

Step 2.3.8.2

Simplify.

$- \cos (y) + 2 \sin (y) + C_{3} = - e^{2} x - \frac{1}{2} x^{2} - x + C_{7}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Use the identity to solve the equation. In this identity, $θ$ represents the angle created by plotting point $(a, b)$ on a graph and therefore can be found using $θ = \arctan (\frac{b}{a})$ .

$a \sin (x) + b \cos (x) = R \sin (x + θ)$ where $R = \sqrt{a^{2} + b^{2}}$ and $θ = \arctan (\frac{b}{a})$

Step 3.2

Set up the equation to find the value of $θ$ .

$\arctan (- \frac{1}{2})$

Step 3.3

Evaluate $\arctan (- \frac{1}{2})$ .

$θ = - 0.4636476$

Step 3.4

Solve to find the value of $R$ .

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Step 3.4.1

Raise $2$ to the power of $2$ .

$R = \sqrt{4 + {(- 1)}^{2}}$

Step 3.4.2

Raise $- 1$ to the power of $2$ .

$R = \sqrt{4 + 1}$

Step 3.4.3

Add $4$ and $1$ .

$R = \sqrt{5}$

Step 3.5

Substitute the known values into the equation.

$(\sqrt{5}) \sin (y - 0.4636476) = - e^{2} x - \frac{1}{2} x^{2} - x + K$

Step 3.6

Divide each term in $\sqrt{5} \sin (y - 0.4636476) = - e^{2} x - \frac{1}{2} x^{2} - x + K$ by $\sqrt{5}$ and simplify.

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Step 3.6.1

Divide each term in $\sqrt{5} \sin (y - 0.4636476) = - e^{2} x - \frac{1}{2} x^{2} - x + K$ by $\sqrt{5}$ .

$\frac{\sqrt{5} \sin (y - 0.4636476)}{\sqrt{5}} = \frac{- e^{2} x}{\sqrt{5}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $\sqrt{5}$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{\sqrt{5} \sin (y - 0.4636476)}{\sqrt{5}} = \frac{- e^{2} x}{\sqrt{5}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.2.1.2

Divide $\sin (y - 0.4636476)$ by $1$ .

$\sin (y - 0.4636476) = \frac{- e^{2} x}{\sqrt{5}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Simplify each term.

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Step 3.6.3.1.1

Move the negative in front of the fraction.

$\sin (y - 0.4636476) = - \frac{e^{2} x}{\sqrt{5}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.2

Multiply $\frac{e^{2} x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y - 0.4636476) = - (\frac{e^{2} x}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}) + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3

Combine and simplify the denominator.

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Step 3.6.3.1.3.1

Multiply $\frac{e^{2} x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{\sqrt{5} \sqrt{5}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{{\sqrt{5}}^{1 + 1}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.5

Add $1$ and $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{{\sqrt{5}}^{2}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.6.3.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.6.3

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

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5^{\frac{2}{2}}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.6.4

Cancel the common factor of $2$ .

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Step 3.6.3.1.3.6.4.1

Cancel the common factor.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5^{\frac{2}{2}}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.6.4.2

Rewrite the expression.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5^{1}} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.3.6.5

Evaluate the exponent.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} + \frac{- \frac{1}{2} x^{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.4

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

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} + \frac{- \frac{x^{2}}{2}}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.5

Multiply the numerator by the reciprocal of the denominator.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{1}{\sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.6

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

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}) + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7

Combine and simplify the denominator.

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Step 3.6.3.1.7.1

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

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.5

Add $1$ and $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{2}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.6.3.1.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.6.3

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

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{5^{\frac{2}{2}}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.6.4

Cancel the common factor of $2$ .

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Step 3.6.3.1.7.6.4.1

Cancel the common factor.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{5^{\frac{2}{2}}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.6.4.2

Rewrite the expression.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{5^{1}} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.7.6.5

Evaluate the exponent.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{5} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.8

Multiply $- \frac{x^{2}}{2} \cdot \frac{\sqrt{5}}{5}$ .

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Step 3.6.3.1.8.1

Multiply $\frac{\sqrt{5}}{5}$ by $\frac{x^{2}}{2}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{5 \cdot 2} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.8.2

Multiply $5$ by $2$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} + \frac{- x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.9

Move the negative in front of the fraction.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x}{\sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.10

Multiply $\frac{x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - (\frac{x}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}) + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11

Combine and simplify the denominator.

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Step 3.6.3.1.11.1

Multiply $\frac{x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{\sqrt{5} \sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{{\sqrt{5}}^{1 + 1}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.5

Add $1$ and $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{{\sqrt{5}}^{2}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.6.3.1.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.6.3

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

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5^{\frac{2}{2}}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.6.4

Cancel the common factor of $2$ .

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Step 3.6.3.1.11.6.4.1

Cancel the common factor.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5^{\frac{2}{2}}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.6.4.2

Rewrite the expression.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5^{1}} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.11.6.5

Evaluate the exponent.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K}{\sqrt{5}}$

Step 3.6.3.1.12

Multiply $\frac{K}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 3.6.3.1.13

Combine and simplify the denominator.

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Step 3.6.3.1.13.1

Multiply $\frac{K}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.6.3.1.13.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.6.3.1.13.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.6.3.1.13.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.6.3.1.13.5

Add $1$ and $1$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.6.3.1.13.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.6.3.1.13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.6.3.1.13.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.6.3.1.13.6.3

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

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.6.3.1.13.6.4

Cancel the common factor of $2$ .

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Step 3.6.3.1.13.6.4.1

Cancel the common factor.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.6.3.1.13.6.4.2

Rewrite the expression.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{5^{1}}$

Step 3.6.3.1.13.6.5

Evaluate the exponent.

$\sin (y - 0.4636476) = - \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{5}$

Step 3.7

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

$y - 0.4636476 = \arcsin (- \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{5})$

Step 3.8

Add $0.4636476$ to both sides of the equation.

$y = \arcsin (- \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + \frac{K \sqrt{5}}{5}) + 0.4636476$

Step 4

Simplify the constant of integration.

$y = \arcsin (- \frac{e^{2} x \sqrt{5}}{5} - \frac{\sqrt{5} x^{2}}{10} - \frac{x \sqrt{5}}{5} + K) + 0.4636476$