Calculus Examples

$\frac{d y}{d x} = 25 + y^{2}$

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

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.

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

Step 3.1.2.3.2

Rewrite the expression.

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

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 $x$ .

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

Step 3.1.3.1.2

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

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

Step 3.2

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

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

Step 3.3

Multiply both sides of the equation by $5$ .

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

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.

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

Step 3.4.1.2

Rewrite the expression.

$y = 5 \tan (5 x + 5 K)$

Step 4

Simplify the constant of integration.

$y = 5 \tan (5 x + K)$