Calculus Examples

$\frac{d x}{d t} = x^{2} + \frac{1}{81}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{x^{2} + \frac{1}{81}}$ .

$\frac{1}{x^{2} + \frac{1}{81}} \frac{d x}{d t} = \frac{1}{x^{2} + \frac{1}{81}} (x^{2} + \frac{1}{81})$

Step 1.2

Cancel the common factor of $x^{2} + \frac{1}{81}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{x^{2} + \frac{1}{81}} \frac{d x}{d t} = \frac{1}{x^{2} + \frac{1}{81}} (x^{2} + \frac{1}{81})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{x^{2} + \frac{1}{81}} \frac{d x}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{x^{2} + \frac{1}{81}} d x = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{x^{2} + \frac{1}{81}} d x = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Reorder $x^{2}$ and $\frac{1}{81}$ .

$\int \frac{1}{\frac{1}{81} + x^{2}} d x = \int d t$

Step 2.2.2

Rewrite $\frac{1}{81}$ as ${(\frac{1}{9})}^{2}$ .

$\int \frac{1}{{(\frac{1}{9})}^{2} + x^{2}} d x = \int d t$

Step 2.2.3

The integral of $\frac{1}{{(\frac{1}{9})}^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{\frac{1}{9}} \arctan (\frac{x}{\frac{1}{9}}) + C_{1}$ .

$\frac{1}{\frac{1}{9}} \arctan (\frac{x}{\frac{1}{9}}) + C_{1} = \int d t$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{9}$ .

$1 \cdot 9 \arctan (\frac{x}{\frac{1}{9}}) + C_{1} = \int d t$

Step 2.2.4.2

Multiply $9$ by $1$ .

$9 \arctan (\frac{x}{\frac{1}{9}}) + C_{1} = \int d t$

Step 2.2.4.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{9}$ .

$9 \arctan (x \cdot 9) + C_{1} = \int d t$

Step 2.2.4.4

Move $9$ to the left of $x$ .

$9 \arctan (9 x) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$9 \arctan (9 x) + C_{1} = t + C_{2}$

Step 2.4

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

$9 \arctan (9 x) = t + K$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $9 \arctan (9 x) = t + K$ by $9$ and simplify.

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Step 3.1.1

Divide each term in $9 \arctan (9 x) = t + K$ by $9$ .

$\frac{9 \arctan (9 x)}{9} = \frac{t}{9} + \frac{K}{9}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $9$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{9 \arctan (9 x)}{9} = \frac{t}{9} + \frac{K}{9}$

Step 3.1.2.1.2

Divide $\arctan (9 x)$ by $1$ .

$\arctan (9 x) = \frac{t}{9} + \frac{K}{9}$

Step 3.2

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

$9 x = \tan (\frac{t}{9} + \frac{K}{9})$

Step 3.3

Divide each term in $9 x = \tan (\frac{t}{9} + \frac{K}{9})$ by $9$ and simplify.

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Step 3.3.1

Divide each term in $9 x = \tan (\frac{t}{9} + \frac{K}{9})$ by $9$ .

$\frac{9 x}{9} = \frac{\tan (\frac{t}{9} + \frac{K}{9})}{9}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $9$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{\tan (\frac{t}{9} + \frac{K}{9})}{9}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\tan (\frac{t}{9} + \frac{K}{9})}{9}$

Step 4

Simplify the constant of integration.

$x = \frac{\tan (\frac{t}{9} + K)}{9}$