Calculus Examples

$\frac{d y}{d x} = 1 + 4 y^{2}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{1 + 4 y^{2}}$ .

$\frac{1}{1 + 4 y^{2}} \frac{d y}{d x} = \frac{1}{1 + 4 y^{2}} (1 + 4 y^{2})$

Step 1.2

Cancel the common factor of $1 + 4 y^{2}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{1 + 4 y^{2}} \frac{d y}{d x} = \frac{1}{1 + 4 y^{2}} (1 + 4 y^{2})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{1 + 4 y^{2}} \frac{d y}{d x} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{1 + 4 y^{2}} d y = 1 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{1 + 4 y^{2}} d y = \int d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Factor $4$ out of $1 + 4 y^{2}$ .

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Step 2.2.1.1

Factor $4$ out of $1$ .

$\int \frac{1}{4 (\frac{1}{4}) + 4 y^{2}} d y = \int d x$

Step 2.2.1.2

Factor $4$ out of $4 y^{2}$ .

$\int \frac{1}{4 (\frac{1}{4}) + 4 (y^{2})} d y = \int d x$

Step 2.2.1.3

Factor $4$ out of $4 (\frac{1}{4}) + 4 (y^{2})$ .

$\int \frac{1}{4 (\frac{1}{4} + y^{2})} d y = \int d x$

Step 2.2.2

Since $\frac{1}{4}$ is constant with respect to $y$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \frac{1}{\frac{1}{4} + y^{2}} d y = \int d x$

Step 2.2.3

Rewrite $\frac{1}{4}$ as ${(\frac{1}{2})}^{2}$ .

$\frac{1}{4} \int \frac{1}{{(\frac{1}{2})}^{2} + y^{2}} d y = \int d x$

Step 2.2.4

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

$\frac{1}{4} (\frac{1}{\frac{1}{2}} \arctan (\frac{y}{\frac{1}{2}}) + C_{1}) = \int d x$

Step 2.2.5

Simplify the answer.

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Step 2.2.5.1

Simplify.

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Step 2.2.5.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{4} (1 \cdot 2 \arctan (\frac{y}{\frac{1}{2}}) + C_{1}) = \int d x$

Step 2.2.5.1.2

Multiply $2$ by $1$ .

$\frac{1}{4} (2 \arctan (\frac{y}{\frac{1}{2}}) + C_{1}) = \int d x$

Step 2.2.5.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{4} (2 \arctan (y \cdot 2) + C_{1}) = \int d x$

Step 2.2.5.1.4

Move $2$ to the left of $y$ .

$\frac{1}{4} (2 \arctan (2 y) + C_{1}) = \int d x$

Step 2.2.5.2

Rewrite $\frac{1}{4} (2 \arctan (2 y) + C_{1})$ as $\frac{1}{4} \cdot 2 \arctan (2 y) + C_{1}$ .

$\frac{1}{4} \cdot 2 \arctan (2 y) + C_{1} = \int d x$

Step 2.2.5.3

Simplify.

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Step 2.2.5.3.1

Combine $\frac{1}{4}$ and $2$ .

$\frac{2}{4} \arctan (2 y) + C_{1} = \int d x$

Step 2.2.5.3.2

Cancel the common factor of $2$ and $4$ .

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Step 2.2.5.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} \arctan (2 y) + C_{1} = \int d x$

Step 2.2.5.3.2.2

Cancel the common factors.

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Step 2.2.5.3.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} \arctan (2 y) + C_{1} = \int d x$

Step 2.2.5.3.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} \arctan (2 y) + C_{1} = \int d x$

Step 2.2.5.3.2.2.3

Rewrite the expression.

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply each term in $\frac{1}{2} \arctan (2 y) = x + K$ by $2$ to eliminate the fractions.

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Step 3.1.1

Multiply each term in $\frac{1}{2} \arctan (2 y) = x + K$ by $2$ .

$\frac{1}{2} \arctan (2 y) \cdot 2 = x \cdot 2 + K \cdot 2$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Combine $\frac{1}{2}$ and $\arctan (2 y)$ .

$\frac{\arctan (2 y)}{2} \cdot 2 = x \cdot 2 + K \cdot 2$

Step 3.1.2.2

Cancel the common factor of $2$ .

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Step 3.1.2.2.1

Cancel the common factor.

$\frac{\arctan (2 y)}{2} \cdot 2 = x \cdot 2 + K \cdot 2$

Step 3.1.2.2.2

Rewrite the expression.

$\arctan (2 y) = x \cdot 2 + K \cdot 2$

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 $2$ to the left of $x$ .

$\arctan (2 y) = 2 \cdot x + K \cdot 2$

Step 3.1.3.1.2

Move $2$ to the left of $K$ .

$\arctan (2 y) = 2 x + 2 K$

Step 3.2

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

$2 y = \tan (2 x + 2 K)$

Step 3.3

Divide each term in $2 y = \tan (2 x + 2 K)$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 y = \tan (2 x + 2 K)$ by $2$ .

$\frac{2 y}{2} = \frac{\tan (2 x + 2 K)}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\tan (2 x + 2 K)}{2}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\tan (2 x + 2 K)}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\tan (2 x + K)}{2}$