Calculus Examples

$\frac{d y}{d x} = \cot (x)$ , $y (- 3) = 7$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

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

$y + C_{1} = \ln (| \sin (x) |) + C_{2}$

Step 2.4

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

$y = \ln (| \sin (x) |) + C$

Step 3

Use the initial condition to find the value of $C$ by substituting $- 3$ for $x$ and $7$ for $y$ in $y = \ln (| \sin (x) |) + C$ .

$7 = \ln (| \sin (- 3) |) + C$

Step 4

Solve for $C$ .

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Step 4.1

Rewrite the equation as $\ln (| \sin (- 3) |) + C = 7$ .

$\ln (| \sin (- 3) |) + C = 7$

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

Evaluate $\sin (- 3)$ .

$\ln (| - 0.05233595 |) + C = 7$

Step 4.2.1.2

The absolute value is the distance between a number and zero. The distance between $- 0.05233595$ and $0$ is $0.05233595$ .

$\ln (0.05233595) + C = 7$

Step 4.3

Subtract $\ln (0.05233595)$ from both sides of the equation.

$C = 7 - \ln (0.05233595)$

Step 5

Substitute $7 - \ln (0.05233595)$ for $C$ in $y = \ln (| \sin (x) |) + C$ and simplify.

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Step 5.1

Substitute $7 - \ln (0.05233595)$ for $C$ .

$y = \ln (| \sin (x) |) + 7 - \ln (0.05233595)$

Step 5.2

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

$y = \ln (\frac{| \sin (x) |}{0.05233595}) + 7$

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply by $1$ .

$y = \ln (\frac{1 | \sin (x) |}{0.05233595}) + 7$

Step 5.3.2

Factor $0.05233595$ out of $0.05233595$ .

$y = \ln (\frac{1 | \sin (x) |}{0.05233595 (1)}) + 7$

Step 5.3.3

Separate fractions.

$y = \ln (\frac{1}{0.05233595} \cdot \frac{| \sin (x) |}{1}) + 7$

Step 5.3.4

Divide $1$ by $0.05233595$ .

$y = \ln (19.1073226 \frac{| \sin (x) |}{1}) + 7$

Step 5.3.5

Divide $| \sin (x) |$ by $1$ .

$y = \ln (19.1073226 | \sin (x) |) + 7$