Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Subtract $e^{- y} \cos (x)$ from both sides of the equation.

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

Step 1.1.2

Divide each term in $\frac{d y}{d x} \sin^{2} (x) = - e^{- y} \cos (x)$ by $\sin^{2} (x)$ and simplify.

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Step 1.1.2.1

Divide each term in $\frac{d y}{d x} \sin^{2} (x) = - e^{- y} \cos (x)$ by $\sin^{2} (x)$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $\sin^{2} (x)$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\frac{d y}{d x} = \frac{- e^{- y} \cos (x)}{\sin (x) \sin (x)}$

Step 1.1.2.3.2

Separate fractions.

$\frac{d y}{d x} = \frac{- e^{- y}}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 1.1.2.3.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{d y}{d x} = \frac{- e^{- y}}{\sin (x)} \cot (x)$

Step 1.1.2.3.4

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{e^{- y}}{\sin (x)} \cot (x)$

Step 1.1.2.3.5

Separate fractions.

$\frac{d y}{d x} = - (\frac{e^{- y}}{1} \cdot \frac{1}{\sin (x)}) \cot (x)$

Step 1.1.2.3.6

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{d y}{d x} = - (\frac{e^{- y}}{1} \csc (x)) \cot (x)$

Step 1.1.2.3.7

Divide $e^{- y}$ by $1$ .

$\frac{d y}{d x} = - e^{- y} \csc (x) \cot (x)$

Step 1.2

Multiply both sides by $\frac{1}{e^{- y}}$ .

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{1}{e^{- y}} (- e^{- y} \csc (x) \cot (x))$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{e^{- y}} \frac{d y}{d x} = - \frac{1}{e^{- y}} (e^{- y} \csc (x) \cot (x))$

Step 1.3.2

Cancel the common factor of $e^{- y}$ .

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Step 1.3.2.1

Move the leading negative in $- \frac{1}{e^{- y}}$ into the numerator.

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{- 1}{e^{- y}} (e^{- y} \csc (x) \cot (x))$

Step 1.3.2.2

Factor $e^{- y}$ out of $e^{- y} \csc (x) \cot (x)$ .

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{- 1}{e^{- y}} (e^{- y} (\csc (x) \cot (x)))$

Step 1.3.2.3

Cancel the common factor.

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{- 1}{e^{- y}} (e^{- y} (\csc (x) \cot (x)))$

Step 1.3.2.4

Rewrite the expression.

$\frac{1}{e^{- y}} \frac{d y}{d x} = - (\csc (x) \cot (x))$

$\frac{1}{e^{- y}} \frac{d y}{d x} = - \csc (x) \cot (x)$

Step 1.4

Rewrite the equation.

$\frac{1}{e^{- y}} d y = - \csc (x) \cot (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{e^{- y}} d y = \int - \csc (x) \cot (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Negate the exponent of $e^{- y}$ and move it out of the denominator.

$\int 1 {(e^{- y})}^{- 1} d y = \int - \csc (x) \cot (x) d x$

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

Multiply the exponents in ${(e^{- y})}^{- 1}$ .

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Step 2.2.1.2.1.1

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

$\int 1 e^{- y \cdot - 1} d y = \int - \csc (x) \cot (x) d x$

Step 2.2.1.2.1.2

Multiply $- y \cdot - 1$ .

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Step 2.2.1.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\int 1 e^{1 y} d y = \int - \csc (x) \cot (x) d x$

Step 2.2.1.2.1.2.2

Multiply $y$ by $1$ .

$\int 1 e^{y} d y = \int - \csc (x) \cot (x) d x$

Step 2.2.1.2.2

Multiply $e^{y}$ by $1$ .

$\int e^{y} d y = \int - \csc (x) \cot (x) d x$

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Since the derivative of $- \csc (x)$ is $\csc (x) \cot (x)$ , the integral of $\csc (x) \cot (x)$ is $- \csc (x)$ .

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Simplify.

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.2.2

Multiply $\csc (x)$ by $1$ .

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

Step 2.4

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

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

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.

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

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.

$y \ln (e) = \ln (\csc (x) + K)$

Step 3.2.2

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

$y \cdot 1 = \ln (\csc (x) + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (\csc (x) + K)$