Calculus Examples

$\frac{d R}{d x} = a (R^{2} + 25)$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{R^{2} + 25}$ .

$\frac{1}{R^{2} + 25} \frac{d R}{d x} = \frac{1}{R^{2} + 25} (a (R^{2} + 25))$

Step 1.2

Cancel the common factor of $R^{2} + 25$ .

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Step 1.2.1

Factor $R^{2} + 25$ out of $a (R^{2} + 25)$ .

$\frac{1}{R^{2} + 25} \frac{d R}{d x} = \frac{1}{R^{2} + 25} ((R^{2} + 25) a)$

Step 1.2.2

Cancel the common factor.

$\frac{1}{R^{2} + 25} \frac{d R}{d x} = \frac{1}{R^{2} + 25} ((R^{2} + 25) a)$

Step 1.2.3

Rewrite the expression.

$\frac{1}{R^{2} + 25} \frac{d R}{d x} = a$

Step 1.3

Rewrite the equation.

$\frac{1}{R^{2} + 25} d R = a d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{R^{2} + 25} d R = \int a d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Reorder $R^{2}$ and $25$ .

$\int \frac{1}{25 + R^{2}} d R = \int a d x$

Step 2.2.1.2

Rewrite $25$ as $5^{2}$ .

$\int \frac{1}{5^{2} + R^{2}} d R = \int a d x$

Step 2.2.2

The integral of $\frac{1}{5^{2} + R^{2}}$ with respect to $R$ is $\frac{1}{5} \arctan (\frac{R}{5}) + C_{1}$ .

$\frac{1}{5} \arctan (\frac{R}{5}) + C_{1} = \int a d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Combine $\frac{1}{5}$ and $\arctan (\frac{R}{5})$ .

$\frac{\arctan (\frac{R}{5})}{5} + C_{1} = \int a d x$

Step 2.2.3.2

Rewrite $\frac{\arctan (\frac{R}{5})}{5} + C_{1}$ as $\frac{1}{5} \arctan (\frac{1}{5} R) + C_{1}$ .

$\frac{1}{5} \arctan (\frac{1}{5} R) + C_{1} = \int a d x$

Step 2.3

Apply the constant rule.

$\frac{1}{5} \arctan (\frac{1}{5} R) + C_{1} = a x + C_{2}$

Step 2.4

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

$\frac{1}{5} \arctan (\frac{1}{5} R) = a x + K$

Step 3

Solve for $R$ .

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Step 3.1

Multiply each term in $\frac{1}{5} \arctan (\frac{1}{5} R) = a x + K$ by $5$ to eliminate the fractions.

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Step 3.1.1

Multiply each term in $\frac{1}{5} \arctan (\frac{1}{5} R) = a x + K$ by $5$ .

$\frac{1}{5} \arctan (\frac{1}{5} R) \cdot 5 = a x \cdot 5 + K \cdot 5$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Combine $\frac{1}{5}$ and $R$ .

$\frac{1}{5} \arctan (\frac{R}{5}) \cdot 5 = a x \cdot 5 + K \cdot 5$

Step 3.1.2.2

Combine $\frac{1}{5}$ and $\arctan (\frac{R}{5})$ .

$\frac{\arctan (\frac{R}{5})}{5} \cdot 5 = a x \cdot 5 + K \cdot 5$

Step 3.1.2.3

Cancel the common factor of $5$ .

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Step 3.1.2.3.1

Cancel the common factor.

$\frac{\arctan (\frac{R}{5})}{5} \cdot 5 = a x \cdot 5 + K \cdot 5$

Step 3.1.2.3.2

Rewrite the expression.

$\arctan (\frac{R}{5}) = a x \cdot 5 + K \cdot 5$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move $5$ to the left of $a x$ .

$\arctan (\frac{R}{5}) = 5 \cdot (a x) + K \cdot 5$

Step 3.1.3.1.2

Move $5$ to the left of $K$ .

$\arctan (\frac{R}{5}) = 5 a x + 5 K$

Step 3.2

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

$\frac{R}{5} = \tan (5 a x + 5 K)$

Step 3.3

Multiply both sides of the equation by $5$ .

$5 \frac{R}{5} = 5 \tan (5 a x + 5 K)$

Step 3.4

Simplify the left side.

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Step 3.4.1

Cancel the common factor of $5$ .

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Step 3.4.1.1

Cancel the common factor.

$5 \frac{R}{5} = 5 \tan (5 a x + 5 K)$

Step 3.4.1.2

Rewrite the expression.

$R = 5 \tan (5 a x + 5 K)$

Step 4

Simplify the constant of integration.

$R = 5 \tan (5 a x + K)$