Calculus Examples

$\frac{d x}{d t} = x^{2} + \frac{1}{36}$ , $x (0) = 2$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{x^{2} + \frac{1}{36}}$ .

$\frac{1}{x^{2} + \frac{1}{36}} \frac{d x}{d t} = \frac{1}{x^{2} + \frac{1}{36}} (x^{2} + \frac{1}{36})$

Step 1.2

Cancel the common factor of $x^{2} + \frac{1}{36}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{x^{2} + \frac{1}{36}} \frac{d x}{d t} = \frac{1}{x^{2} + \frac{1}{36}} (x^{2} + \frac{1}{36})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{x^{2} + \frac{1}{36}} \frac{d x}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{x^{2} + \frac{1}{36}} d x = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{x^{2} + \frac{1}{36}} d x = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Reorder $x^{2}$ and $\frac{1}{36}$ .

$\int \frac{1}{\frac{1}{36} + x^{2}} d x = \int d t$

Step 2.2.2

Rewrite $\frac{1}{36}$ as ${(\frac{1}{6})}^{2}$ .

$\int \frac{1}{{(\frac{1}{6})}^{2} + x^{2}} d x = \int d t$

Step 2.2.3

The integral of $\frac{1}{{(\frac{1}{6})}^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{\frac{1}{6}} \arctan (\frac{x}{\frac{1}{6}}) + C_{1}$ .

$\frac{1}{\frac{1}{6}} \arctan (\frac{x}{\frac{1}{6}}) + C_{1} = \int d t$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{6}$ .

$1 \cdot 6 \arctan (\frac{x}{\frac{1}{6}}) + C_{1} = \int d t$

Step 2.2.4.2

Multiply $6$ by $1$ .

$6 \arctan (\frac{x}{\frac{1}{6}}) + C_{1} = \int d t$

Step 2.2.4.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{6}$ .

$6 \arctan (x \cdot 6) + C_{1} = \int d t$

Step 2.2.4.4

Move $6$ to the left of $x$ .

$6 \arctan (6 x) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$6 \arctan (6 x) + C_{1} = t + C_{2}$

Step 2.4

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

$6 \arctan (6 x) = t + K$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $6 \arctan (6 x) = t + K$ by $6$ and simplify.

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Step 3.1.1

Divide each term in $6 \arctan (6 x) = t + K$ by $6$ .

$\frac{6 \arctan (6 x)}{6} = \frac{t}{6} + \frac{K}{6}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $6$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{6 \arctan (6 x)}{6} = \frac{t}{6} + \frac{K}{6}$

Step 3.1.2.1.2

Divide $\arctan (6 x)$ by $1$ .

$\arctan (6 x) = \frac{t}{6} + \frac{K}{6}$

Step 3.2

Take the inverse arctangent of both sides of the equation to extract $x$ from inside the arctangent.

$6 x = \tan (\frac{t}{6} + \frac{K}{6})$

Step 3.3

Divide each term in $6 x = \tan (\frac{t}{6} + \frac{K}{6})$ by $6$ and simplify.

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Step 3.3.1

Divide each term in $6 x = \tan (\frac{t}{6} + \frac{K}{6})$ by $6$ .

$\frac{6 x}{6} = \frac{\tan (\frac{t}{6} + \frac{K}{6})}{6}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\tan (\frac{t}{6} + \frac{K}{6})}{6}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\tan (\frac{t}{6} + \frac{K}{6})}{6}$

Step 4

Simplify the constant of integration.

$x = \frac{\tan (\frac{t}{6} + K)}{6}$

Step 5

Use the initial condition to find the value of $K$ by substituting $0$ for $t$ and $2$ for $x$ in $x = \frac{\tan (\frac{t}{6} + K)}{6}$ .

$2 = \frac{\tan (\frac{0}{6} + K)}{6}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\frac{\tan (\frac{0}{6} + K)}{6} = 2$ .

$\frac{\tan (\frac{0}{6} + K)}{6} = 2$

Step 6.2

Multiply both sides of the equation by $6$ .

$6 \frac{\tan (\frac{0}{6} + K)}{6} = 6 \cdot 2$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Cancel the common factor of $6$ .

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Step 6.3.1.1.1

Cancel the common factor.

$6 \frac{\tan (\frac{0}{6} + K)}{6} = 6 \cdot 2$

Step 6.3.1.1.2

Rewrite the expression.

$\tan (\frac{0}{6} + K) = 6 \cdot 2$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Multiply $6$ by $2$ .

$\tan (\frac{0}{6} + K) = 12$

Step 6.4

Take the inverse tangent of both sides of the equation to extract $K$ from inside the tangent.

$\frac{0}{6} + K = \arctan (12)$

Step 6.5

Simplify the left side.

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Step 6.5.1

Combine the opposite terms in $\frac{0}{6} + K$ .

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Step 6.5.1.1

Divide $0$ by $6$ .

$0 + K = \arctan (12)$

Step 6.5.1.2

Add $0$ and $K$ .

$K = \arctan (12)$

Step 6.6

Simplify the right side.

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Step 6.6.1

Evaluate $\arctan (12)$ .

$K = 1.48765509$

Step 6.7

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$K = (3.14159265) + 1.48765509$

Step 6.8

Solve for $K$ .

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Step 6.8.1

Remove parentheses.

$K = 3.14159265 + 1.48765509$

Step 6.8.2

Remove parentheses.

$K = (3.14159265) + 1.48765509$

Step 6.8.3

Add $3.14159265$ and $1.48765509$ .

$K = 4.62924774$

Step 6.9

Find the period of $\tan (\frac{0}{6} + K)$ .

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Step 6.9.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 6.9.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 6.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 6.9.4

Divide $π$ by $1$ .

$π$

Step 6.10

The period of the $\tan (\frac{0}{6} + K)$ function is $π$ so values will repeat every $π$ radians in both directions.

$K = 1.48765509 + π n, 4.62924774 + π n$ , for any integer $n$

Step 6.11

Consolidate $1.48765509 + π n$ and $4.62924774 + π n$ to $1.48765509 + π n$ .

$K = 1.48765509 + π n$

Step 7

Substitute $1.48765509 + π n$ for $K$ in $x = \frac{\tan (\frac{t}{6} + K)}{6}$ and simplify.

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Step 7.1

Substitute $1.48765509 + π n$ for $K$ .

$x = \frac{\tan (\frac{t}{6} + 1.48765509 + π n)}{6}$