Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Factor $x$ out of $4 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

Factor $x$ out of $x \cdot 4 + x (y^{2})$ .

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

Step 1.1.4

Multiply $x$ by $4 + y^{2}$ .

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

Step 1.2

Regroup factors.

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 2 + x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 2 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $2 + x^{2}$ .

$\frac{d}{d x} [2 + x^{2}]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $2 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [x^{2}]$

Step 2.3.1.1.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 2.3.1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 + 2 x$

Step 2.3.1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 2.3.1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

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

Step 2.3.2.2

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

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

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.4

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{1}{2} \arctan (\frac{1}{2} y) + C_{1} = \frac{1}{2} (\ln (| u |) + C_{2})$

Step 2.3.5

Simplify.

$\frac{1}{2} \arctan (\frac{1}{2} y) + C_{1} = \frac{1}{2} \ln (| u |) + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $2 + x^{2}$ .

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

Step 2.4

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

$\frac{1}{2} \arctan (\frac{1}{2} y) = \frac{1}{2} \ln (| 2 + x^{2} |) + C$

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

$\frac{1}{2} \arctan (\frac{1}{2} y) \cdot 2 = \frac{1}{2} \ln (| 2 + x^{2} |) \cdot 2 + C \cdot 2$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

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

$\frac{1}{2} \arctan (\frac{y}{2}) \cdot 2 = \frac{1}{2} \ln (| 2 + x^{2} |) \cdot 2 + C \cdot 2$

Step 3.1.2.2

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

$\frac{\arctan (\frac{y}{2})}{2} \cdot 2 = \frac{1}{2} \ln (| 2 + x^{2} |) \cdot 2 + C \cdot 2$

Step 3.1.2.3

Cancel the common factor of $2$ .

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Step 3.1.2.3.1

Cancel the common factor.

$\frac{\arctan (\frac{y}{2})}{2} \cdot 2 = \frac{1}{2} \ln (| 2 + x^{2} |) \cdot 2 + C \cdot 2$

Step 3.1.2.3.2

Rewrite the expression.

$\arctan (\frac{y}{2}) = \frac{1}{2} \ln (| 2 + x^{2} |) \cdot 2 + C \cdot 2$

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

Simplify $\frac{1}{2} \ln (| 2 + x^{2} |)$ by moving $\frac{1}{2}$ inside the logarithm.

$\arctan (\frac{y}{2}) = \ln ({| 2 + x^{2} |}^{\frac{1}{2}}) \cdot 2 + C \cdot 2$

Step 3.1.3.1.2

Multiply $\ln ({| 2 + x^{2} |}^{\frac{1}{2}}) \cdot 2$ .

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Step 3.1.3.1.2.1

Reorder $\ln ({| 2 + x^{2} |}^{\frac{1}{2}})$ and $2$ .

$\arctan (\frac{y}{2}) = 2 \cdot \ln ({| 2 + x^{2} |}^{\frac{1}{2}}) + C \cdot 2$

Step 3.1.3.1.2.2

Simplify $2 \cdot \ln ({| 2 + x^{2} |}^{\frac{1}{2}})$ by moving $2$ inside the logarithm.

$\arctan (\frac{y}{2}) = \ln ({({| 2 + x^{2} |}^{\frac{1}{2}})}^{2}) + C \cdot 2$

Step 3.1.3.1.3

Multiply the exponents in ${({| 2 + x^{2} |}^{\frac{1}{2}})}^{2}$ .

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Step 3.1.3.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\arctan (\frac{y}{2}) = \ln ({| 2 + x^{2} |}^{\frac{1}{2} \cdot 2}) + C \cdot 2$

Step 3.1.3.1.3.2

Cancel the common factor of $2$ .

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Step 3.1.3.1.3.2.1

Cancel the common factor.

$\arctan (\frac{y}{2}) = \ln ({| 2 + x^{2} |}^{\frac{1}{2} \cdot 2}) + C \cdot 2$

Step 3.1.3.1.3.2.2

Rewrite the expression.

$\arctan (\frac{y}{2}) = \ln ({| 2 + x^{2} |}^{1}) + C \cdot 2$

Step 3.1.3.1.4

Simplify.

$\arctan (\frac{y}{2}) = \ln (| 2 + x^{2} |) + C \cdot 2$

Step 3.1.3.1.5

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

$\arctan (\frac{y}{2}) = \ln (| 2 + x^{2} |) + 2 C$

Step 3.2

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

$\frac{y}{2} = \tan (\ln (| 2 + x^{2} |) + 2 C)$

Step 3.3

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 \tan (\ln (| 2 + x^{2} |) + 2 C)$

Step 3.4

Simplify the left side.

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Step 3.4.1

Cancel the common factor of $2$ .

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Step 3.4.1.1

Cancel the common factor.

$2 \frac{y}{2} = 2 \tan (\ln (| 2 + x^{2} |) + 2 C)$

Step 3.4.1.2

Rewrite the expression.

$y = 2 \tan (\ln (| 2 + x^{2} |) + 2 C)$

Step 4

Simplify the constant of integration.

$y = 2 \tan (\ln (| 2 + x^{2} |) + C)$