Calculus Examples

$x^{2} (y + 1) d y - (x + 1) (y d) x = 0$

Step 1

Add $(x + 1) y d x$ to both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $\frac{1}{y}$ by $y + 1$ .

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

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

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

Step 3.3.2

Factor $y$ out of $(x + 1) y$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

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

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

Split the fraction into multiple fractions.

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

Step 4.2.2

Split the single integral into multiple integrals.

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

Step 4.2.3

Cancel the common factor of $y$ .

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Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Rewrite the expression.

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

Step 4.2.4

Apply the constant rule.

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

Step 4.2.5

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

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

Step 4.2.6

Simplify.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Apply basic rules of exponents.

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Step 4.3.1.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$y + \ln (| y |) + C_{3} = \int (x + 1) {(x^{2})}^{- 1} d x$

Step 4.3.1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.3.1.2.1

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

$y + \ln (| y |) + C_{3} = \int (x + 1) x^{2 \cdot - 1} d x$

Step 4.3.1.2.2

Multiply $2$ by $- 1$ .

$y + \ln (| y |) + C_{3} = \int (x + 1) x^{- 2} d x$

Step 4.3.2

Multiply $(x + 1) x^{- 2}$ .

$y + \ln (| y |) + C_{3} = \int x \cdot x^{- 2} + 1 x^{- 2} d x$

Step 4.3.3

Simplify.

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Step 4.3.3.1

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 4.3.3.1.1

Multiply $x$ by $x^{- 2}$ .

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Step 4.3.3.1.1.1

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

$y + \ln (| y |) + C_{3} = \int x^{1} x^{- 2} + 1 x^{- 2} d x$

Step 4.3.3.1.1.2

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

$y + \ln (| y |) + C_{3} = \int x^{1 - 2} + 1 x^{- 2} d x$

Step 4.3.3.1.2

Subtract $2$ from $1$ .

$y + \ln (| y |) + C_{3} = \int x^{- 1} + 1 x^{- 2} d x$

Step 4.3.3.2

Multiply $x^{- 2}$ by $1$ .

$y + \ln (| y |) + C_{3} = \int x^{- 1} + x^{- 2} d x$

Step 4.3.4

Split the single integral into multiple integrals.

$y + \ln (| y |) + C_{3} = \int x^{- 1} d x + \int x^{- 2} d x$

Step 4.3.5

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

$y + \ln (| y |) + C_{3} = \ln (| x |) + C_{4} + \int x^{- 2} d x$

Step 4.3.6

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

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

Step 4.3.7

Simplify.

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

Step 4.3.8

Reorder terms.

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

Step 4.4

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

$y + \ln (| y |) = - \frac{1}{x} + \ln (| x |) + C$