Calculus Examples

$\frac{d y}{d x} = \csc (x)$ , $y (1) = - 6$

Step 1

Rewrite the equation.

$d y = \csc (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \csc (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \csc (x) d x$

Step 2.3

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

$y + C_{1} = \ln (| \csc (x) - \cot (x) |) + C_{2}$

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $C$ by substituting $1$ for $x$ and $- 6$ for $y$ in $y = \ln (| \csc (x) - \cot (x) |) + C$ .

$- 6 = \ln (| \csc (1) - \cot (1) |) + C$

Step 4

Solve for $C$ .

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Step 4.1

Rewrite the equation as $\ln (| \csc (1) - \cot (1) |) + C = - 6$ .

$\ln (| \csc (1) - \cot (1) |) + C = - 6$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Evaluate $\csc (1)$ .

$\ln (| 57.29868849 - \cot (1) |) + C = - 6$

Step 4.2.1.1.2

Evaluate $\cot (1)$ .

$\ln (| 57.29868849 - 1 \cdot 57.28996163 |) + C = - 6$

Step 4.2.1.1.3

Multiply $- 1$ by $57.28996163$ .

$\ln (| 57.29868849 - 57.28996163 |) + C = - 6$

Step 4.2.1.2

Subtract $57.28996163$ from $57.29868849$ .

$\ln (| 0.00872686 |) + C = - 6$

Step 4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.00872686$ is $0.00872686$ .

$\ln (0.00872686) + C = - 6$

Step 4.3

Subtract $\ln (0.00872686)$ from both sides of the equation.

$C = - 6 - \ln (0.00872686)$

Step 5

Substitute $- 6 - \ln (0.00872686)$ for $C$ in $y = \ln (| \csc (x) - \cot (x) |) + C$ and simplify.

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Step 5.1

Substitute $- 6 - \ln (0.00872686)$ for $C$ .

$y = \ln (| \csc (x) - \cot (x) |) - 6 - \ln (0.00872686)$

Step 5.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$y = \ln (\frac{| \csc (x) - \cot (x) |}{0.00872686}) - 6$

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply by $1$ .

$y = \ln (\frac{1 | \csc (x) - \cot (x) |}{0.00872686}) - 6$

Step 5.3.2

Factor $0.00872686$ out of $0.00872686$ .

$y = \ln (\frac{1 | \csc (x) - \cot (x) |}{0.00872686 (1)}) - 6$

Step 5.3.3

Separate fractions.

$y = \ln (\frac{1}{0.00872686} \cdot \frac{| \csc (x) - \cot (x) |}{1}) - 6$

Step 5.3.4

Divide $1$ by $0.00872686$ .

$y = \ln (114.58865012 \frac{| \csc (x) - \cot (x) |}{1}) - 6$

Step 5.3.5

Divide $| \csc (x) - \cot (x) |$ by $1$ .

$y = \ln (114.58865012 | \csc (x) - \cot (x) |) - 6$