Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x y \frac{d y}{d x} = \frac{x^{2} + 1}{y + 1}$ by $x y$ and simplify.

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Step 1.1.1

Divide each term in $x y \frac{d y}{d x} = \frac{x^{2} + 1}{y + 1}$ by $x y$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $y$ .

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.3.2

Multiply $\frac{x^{2} + 1}{y + 1}$ by $\frac{1}{x y}$ .

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

Step 1.1.3.3

Reorder factors in $\frac{x^{2} + 1}{(y + 1) x y}$ .

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

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $y (y + 1)$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

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

Step 1.4.2

Multiply $y$ by $y$ .

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

Step 1.4.3

Multiply $y$ by $1$ .

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

Step 1.4.4

Multiply $\frac{x^{2} + 1}{x}$ by $\frac{1}{y (y + 1)}$ .

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

Step 1.4.5

Multiply $y^{2} + y$ by $\frac{x^{2} + 1}{x y (y + 1)}$ .

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

Step 1.4.6

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

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Step 1.4.6.1

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

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

Step 1.4.6.2

Raise $y$ to the power of $1$ .

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

Step 1.4.6.3

Factor $y$ out of $y^{1}$ .

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

Step 1.4.6.4

Factor $y$ out of $y \cdot y + y \cdot 1$ .

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

Step 1.4.7

Cancel the common factor of $y$ .

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Step 1.4.7.1

Cancel the common factor.

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

Step 1.4.7.2

Rewrite the expression.

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

Step 1.4.8

Cancel the common factor of $y + 1$ .

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Step 1.4.8.1

Cancel the common factor.

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

Step 1.4.8.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Multiply $y (y + 1)$ .

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Raise $y$ to the power of $1$ .

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

Step 2.2.2.2

Raise $y$ to the power of $1$ .

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

Step 2.2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.2.4

Add $1$ and $1$ .

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

Step 2.2.2.5

Multiply $y$ by $1$ .

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

Step 2.2.3

Split the single integral into multiple integrals.

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

Step 2.2.4

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$\frac{1}{3} y^{3} + C_{1} + \int y d y = \int \frac{x^{2} + 1}{x} d x$

Step 2.2.5

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{3} y^{3} + C_{1} + \frac{1}{2} y^{2} + C_{2} = \int \frac{x^{2} + 1}{x} d x$

Step 2.2.6

Simplify.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x^{2} + 1}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x^{2}}{x} + \frac{1}{x} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x^{2}}{x} d x + \int \frac{1}{x} d x$

Step 2.3.3

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.3.3.1

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

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x \cdot x}{x} d x + \int \frac{1}{x} d x$

Step 2.3.3.2

Cancel the common factors.

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Step 2.3.3.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x \cdot x}{x^{1}} d x + \int \frac{1}{x} d x$

Step 2.3.3.2.2

Factor $x$ out of $x^{1}$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x \cdot x}{x \cdot 1} d x + \int \frac{1}{x} d x$

Step 2.3.3.2.3

Cancel the common factor.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x \cdot x}{x \cdot 1} d x + \int \frac{1}{x} d x$

Step 2.3.3.2.4

Rewrite the expression.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int \frac{x}{1} d x + \int \frac{1}{x} d x$

Step 2.3.3.2.5

Divide $x$ by $1$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int x d x + \int \frac{1}{x} d x$

Step 2.3.4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \frac{1}{2} x^{2} + C_{4} + \int \frac{1}{x} d x$

Step 2.3.5

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

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \frac{1}{2} x^{2} + C_{4} + \ln (| x |) + C_{5}$

Step 2.3.6

Simplify.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \frac{1}{2} x^{2} + \ln (| x |) + C_{6}$

Step 2.4

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

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} = \frac{1}{2} x^{2} + \ln (| x |) + C$