Calculus Examples

$\frac{d y}{d x} = \csc (x) + y \cot (x)$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Subtract $y \cot (x)$ from both sides of the equation.

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

Step 1.2

Factor $y$ out of $- y \cot (x)$ .

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

Step 1.3

Reorder $y$ and $- 1 \cot (x)$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 1 \cot (x)$ .

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $- 1 \cot (x)$ .

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Step 2.2.1

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

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

Step 2.2.2

The integral of $\cot (x)$ with respect to $x$ is $\ln (| \sin (x) |)$ .

$e^{- (\ln (| \sin (x) |) + C)}$

Step 2.2.3

Simplify.

$e^{- \ln (| \sin (x) |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- \ln (\sin (x))}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (\sin^{- 1} (x))}$

Step 2.5

Exponentiation and log are inverse functions.

$\sin^{- 1} (x)$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\sin (x)}$

Step 2.7

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

$\csc (x)$

Step 3

Multiply each term by the integrating factor $\csc (x)$ .

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Step 3.1

Multiply each term by $\csc (x)$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

Raise $\csc (x)$ to the power of $1$ .

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

Step 3.3.2

Raise $\csc (x)$ to the power of $1$ .

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

Step 3.3.3

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

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

Step 3.3.4

Add $1$ and $1$ .

$\csc (x) \frac{d y}{d x} - \csc (x) \cot (x) y = \csc^{2} (x)$

Step 3.4

Reorder factors in $\csc (x) \frac{d y}{d x} - \csc (x) \cot (x) y = \csc^{2} (x)$ .

$\csc (x) \frac{d y}{d x} - y \csc (x) \cot (x) = \csc^{2} (x)$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [\csc (x) y] = \csc^{2} (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\csc (x) y] d x = \int \csc^{2} (x) d x$

Step 6

Integrate the left side.

$\csc (x) y = \int \csc^{2} (x) d x$

Step 7

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$\csc (x) y = - \cot (x) + C$

Step 8

Divide each term in $\csc (x) y = - \cot (x) + C$ by $\csc (x)$ and simplify.

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Step 8.1

Divide each term in $\csc (x) y = - \cot (x) + C$ by $\csc (x)$ .

$\frac{\csc (x) y}{\csc (x)} = \frac{- \cot (x)}{\csc (x)} + \frac{C}{\csc (x)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\csc (x)$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{\csc (x) y}{\csc (x)} = \frac{- \cot (x)}{\csc (x)} + \frac{C}{\csc (x)}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \cot (x)}{\csc (x)} + \frac{C}{\csc (x)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Separate fractions.

$y = \frac{- 1}{1} \cdot \frac{\cot (x)}{\csc (x)} + \frac{C}{\csc (x)}$

Step 8.3.1.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$y = \frac{- 1}{1} \cdot \frac{\cot (x)}{\frac{1}{\sin (x)}} + \frac{C}{\csc (x)}$

Step 8.3.1.3

Rewrite $\cot (x)$ in terms of sines and cosines.

$y = \frac{- 1}{1} \cdot \frac{\frac{\cos (x)}{\sin (x)}}{\frac{1}{\sin (x)}} + \frac{C}{\csc (x)}$

Step 8.3.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$y = \frac{- 1}{1} (\frac{\cos (x)}{\sin (x)} \sin (x)) + \frac{C}{\csc (x)}$

Step 8.3.1.5

Write $\sin (x)$ as a fraction with denominator $1$ .

$y = \frac{- 1}{1} (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{1}) + \frac{C}{\csc (x)}$

Step 8.3.1.6

Cancel the common factor of $\sin (x)$ .

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Step 8.3.1.6.1

Cancel the common factor.

$y = \frac{- 1}{1} (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{1}) + \frac{C}{\csc (x)}$

Step 8.3.1.6.2

Rewrite the expression.

$y = \frac{- 1}{1} \cos (x) + \frac{C}{\csc (x)}$

Step 8.3.1.7

Divide $- 1$ by $1$ .

$y = - \cos (x) + \frac{C}{\csc (x)}$

Step 8.3.1.8

Separate fractions.

$y = - \cos (x) + \frac{C}{1} \cdot \frac{1}{\csc (x)}$

Step 8.3.1.9

Rewrite $\csc (x)$ in terms of sines and cosines.

$y = - \cos (x) + \frac{C}{1} \cdot \frac{1}{\frac{1}{\sin (x)}}$

Step 8.3.1.10

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$y = - \cos (x) + \frac{C}{1} (1 \sin (x))$

Step 8.3.1.11

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

$y = - \cos (x) + \frac{C}{1} \sin (x)$

Step 8.3.1.12

Divide $C$ by $1$ .

$y = - \cos (x) + C \sin (x)$