Calculus Examples

$(2 x + 3) y^{6} d x + x^{4} (4 y + 5) d y = 0$

Step 1

Subtract $(2 x + 3) y^{6} d x$ from both sides of the equation.

$x^{4} (4 y + 5) d y = - (2 x + 3) y^{6} d x$

Step 2

Multiply both sides by $\frac{1}{x^{4} y^{6}}$ .

$\frac{1}{x^{4} y^{6}} (x^{4} (4 y + 5)) d y = \frac{1}{x^{4} y^{6}} (- (2 x + 3) y^{6}) d x$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x^{4}$ .

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Step 3.1.1

Factor $x^{4}$ out of $x^{4} y^{6}$ .

$\frac{1}{x^{4} (y^{6})} (x^{4} (4 y + 5)) d y = \frac{1}{x^{4} y^{6}} (- (2 x + 3) y^{6}) d x$

Step 3.1.2

Cancel the common factor.

$\frac{1}{x^{4} y^{6}} (x^{4} (4 y + 5)) d y = \frac{1}{x^{4} y^{6}} (- (2 x + 3) y^{6}) d x$

Step 3.1.3

Rewrite the expression.

$\frac{1}{y^{6}} (4 y + 5) d y = \frac{1}{x^{4} y^{6}} (- (2 x + 3) y^{6}) d x$

Step 3.2

Multiply $\frac{1}{y^{6}}$ by $4 y + 5$ .

$\frac{4 y + 5}{y^{6}} d y = \frac{1}{x^{4} y^{6}} (- (2 x + 3) y^{6}) d x$

Step 3.3

Rewrite using the commutative property of multiplication.

$\frac{4 y + 5}{y^{6}} d y = - \frac{1}{x^{4} y^{6}} ((2 x + 3) y^{6}) d x$

Step 3.4

Cancel the common factor of $y^{6}$ .

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Step 3.4.1

Move the leading negative in $- \frac{1}{x^{4} y^{6}}$ into the numerator.

$\frac{4 y + 5}{y^{6}} d y = \frac{- 1}{x^{4} y^{6}} ((2 x + 3) y^{6}) d x$

Step 3.4.2

Factor $y^{6}$ out of $x^{4} y^{6}$ .

$\frac{4 y + 5}{y^{6}} d y = \frac{- 1}{y^{6} x^{4}} ((2 x + 3) y^{6}) d x$

Step 3.4.3

Factor $y^{6}$ out of $(2 x + 3) y^{6}$ .

$\frac{4 y + 5}{y^{6}} d y = \frac{- 1}{y^{6} x^{4}} (y^{6} (2 x + 3)) d x$

Step 3.4.4

Cancel the common factor.

$\frac{4 y + 5}{y^{6}} d y = \frac{- 1}{y^{6} x^{4}} (y^{6} (2 x + 3)) d x$

Step 3.4.5

Rewrite the expression.

$\frac{4 y + 5}{y^{6}} d y = \frac{- 1}{x^{4}} (2 x + 3) d x$

Step 3.5

Move the negative in front of the fraction.

$\frac{4 y + 5}{y^{6}} d y = - \frac{1}{x^{4}} (2 x + 3) d x$

Step 3.6

Apply the distributive property.

$\frac{4 y + 5}{y^{6}} d y = (- \frac{1}{x^{4}} (2 x) - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.7

Cancel the common factor of $x$ .

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Step 3.7.1

Move the leading negative in $- \frac{1}{x^{4}}$ into the numerator.

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 1}{x^{4}} (2 x) - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.7.2

Factor $x$ out of $x^{4}$ .

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 1}{x \cdot x^{3}} (2 x) - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.7.3

Factor $x$ out of $2 x$ .

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 1}{x \cdot x^{3}} (x \cdot 2) - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.7.4

Cancel the common factor.

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 1}{x \cdot x^{3}} (x \cdot 2) - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.7.5

Rewrite the expression.

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 1}{x^{3}} \cdot 2 - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.8

Combine $\frac{- 1}{x^{3}}$ and $2$ .

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 1 \cdot 2}{x^{3}} - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.9

Multiply $- 1$ by $2$ .

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 2}{x^{3}} - \frac{1}{x^{4}} \cdot 3) d x$

Step 3.10

Multiply $- \frac{1}{x^{4}} \cdot 3$ .

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Step 3.10.1

Multiply $3$ by $- 1$ .

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 2}{x^{3}} - 3 \frac{1}{x^{4}}) d x$

Step 3.10.2

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$\frac{4 y + 5}{y^{6}} d y = (\frac{- 2}{x^{3}} + \frac{- 3}{x^{4}}) d x$

Step 3.11

Simplify each term.

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Step 3.11.1

Move the negative in front of the fraction.

$\frac{4 y + 5}{y^{6}} d y = (- \frac{2}{x^{3}} + \frac{- 3}{x^{4}}) d x$

Step 3.11.2

Move the negative in front of the fraction.

$\frac{4 y + 5}{y^{6}} d y = (- \frac{2}{x^{3}} - \frac{3}{x^{4}}) d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{4 y + 5}{y^{6}} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Apply basic rules of exponents.

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Step 4.2.1.1

Move $y^{6}$ out of the denominator by raising it to the $- 1$ power.

$\int (4 y + 5) {(y^{6})}^{- 1} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.1.2

Multiply the exponents in ${(y^{6})}^{- 1}$ .

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Step 4.2.1.2.1

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

$\int (4 y + 5) y^{6 \cdot - 1} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.1.2.2

Multiply $6$ by $- 1$ .

$\int (4 y + 5) y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.2

Multiply $(4 y + 5) y^{- 6}$ .

$\int 4 y \cdot y^{- 6} + 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.3

Multiply $y$ by $y^{- 6}$ by adding the exponents.

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Step 4.2.3.1

Move $y^{- 6}$ .

$\int 4 (y^{- 6} y) + 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.3.2

Multiply $y^{- 6}$ by $y$ .

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Step 4.2.3.2.1

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

$\int 4 (y^{- 6} y^{1}) + 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.3.2.2

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

$\int 4 y^{- 6 + 1} + 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.3.3

Add $- 6$ and $1$ .

$\int 4 y^{- 5} + 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.4

Split the single integral into multiple integrals.

$\int 4 y^{- 5} d y + \int 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.5

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$4 \int y^{- 5} d y + \int 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.6

By the Power Rule, the integral of $y^{- 5}$ with respect to $y$ is $- \frac{1}{4} y^{- 4}$ .

$4 (- \frac{1}{4} y^{- 4} + C_{1}) + \int 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.7

Simplify.

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Step 4.2.7.1

Combine $y^{- 4}$ and $\frac{1}{4}$ .

$4 (- \frac{y^{- 4}}{4} + C_{1}) + \int 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.7.2

Move $y^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (- \frac{1}{4 y^{4}} + C_{1}) + \int 5 y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.8

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

$4 (- \frac{1}{4 y^{4}} + C_{1}) + 5 \int y^{- 6} d y = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.9

By the Power Rule, the integral of $y^{- 6}$ with respect to $y$ is $- \frac{1}{5} y^{- 5}$ .

$4 (- \frac{1}{4 y^{4}} + C_{1}) + 5 (- \frac{1}{5} y^{- 5} + C_{2}) = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10

Simplify.

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Step 4.2.10.1

Simplify.

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Step 4.2.10.1.1

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

$4 (- \frac{1}{4 y^{4}} + C_{1}) + 5 (- \frac{y^{- 5}}{5} + C_{2}) = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.1.2

Move $y^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (- \frac{1}{4 y^{4}} + C_{1}) + 5 (- \frac{1}{5 y^{5}} + C_{2}) = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.2

Simplify.

$\frac{- 1}{y^{4}} + 5 (- \frac{1}{5 y^{5}}) + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3

Simplify.

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Step 4.2.10.3.1

Move the negative in front of the fraction.

$- \frac{1}{y^{4}} + 5 (- \frac{1}{5 y^{5}}) + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3.2

Multiply $- 1$ by $5$ .

$- \frac{1}{y^{4}} - 5 \frac{1}{5 y^{5}} + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3.3

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

$- \frac{1}{y^{4}} + \frac{- 5}{5 y^{5}} + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3.4

Cancel the common factor of $- 5$ and $5$ .

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Step 4.2.10.3.4.1

Factor $5$ out of $- 5$ .

$- \frac{1}{y^{4}} + \frac{5 \cdot - 1}{5 y^{5}} + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3.4.2

Cancel the common factors.

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Step 4.2.10.3.4.2.1

Factor $5$ out of $5 y^{5}$ .

$- \frac{1}{y^{4}} + \frac{5 \cdot - 1}{5 (y^{5})} + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3.4.2.2

Cancel the common factor.

$- \frac{1}{y^{4}} + \frac{5 \cdot - 1}{5 y^{5}} + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3.4.2.3

Rewrite the expression.

$- \frac{1}{y^{4}} + \frac{- 1}{y^{5}} + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.2.10.3.5

Move the negative in front of the fraction.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = \int - \frac{2}{x^{3}} - \frac{3}{x^{4}} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

Split the single integral into multiple integrals.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = \int - \frac{2}{x^{3}} d x + \int - \frac{3}{x^{4}} d x$

Step 4.3.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - \int \frac{2}{x^{3}} d x + \int - \frac{3}{x^{4}} d x$

Step 4.3.3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - (2 \int \frac{1}{x^{3}} d x) + \int - \frac{3}{x^{4}} d x$

Step 4.3.4

Simplify the expression.

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Step 4.3.4.1

Multiply $2$ by $- 1$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 \int \frac{1}{x^{3}} d x + \int - \frac{3}{x^{4}} d x$

Step 4.3.4.2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 \int {(x^{3})}^{- 1} d x + \int - \frac{3}{x^{4}} d x$

Step 4.3.4.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 4.3.4.3.1

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

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 \int x^{3 \cdot - 1} d x + \int - \frac{3}{x^{4}} d x$

Step 4.3.4.3.2

Multiply $3$ by $- 1$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 \int x^{- 3} d x + \int - \frac{3}{x^{4}} d x$

Step 4.3.5

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

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2} x^{- 2} + C_{4}) + \int - \frac{3}{x^{4}} d x$

Step 4.3.6

Simplify.

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Step 4.3.6.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{x^{- 2}}{2} + C_{4}) + \int - \frac{3}{x^{4}} d x$

Step 4.3.6.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2 x^{2}} + C_{4}) + \int - \frac{3}{x^{4}} d x$

Step 4.3.7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2 x^{2}} + C_{4}) - \int \frac{3}{x^{4}} d x$

Step 4.3.8

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2 x^{2}} + C_{4}) - (3 \int \frac{1}{x^{4}} d x)$

Step 4.3.9

Simplify the expression.

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Step 4.3.9.1

Multiply $3$ by $- 1$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2 x^{2}} + C_{4}) - 3 \int \frac{1}{x^{4}} d x$

Step 4.3.9.2

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2 x^{2}} + C_{4}) - 3 \int {(x^{4})}^{- 1} d x$

Step 4.3.9.3

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 4.3.9.3.1

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

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2 x^{2}} + C_{4}) - 3 \int x^{4 \cdot - 1} d x$

Step 4.3.9.3.2

Multiply $4$ by $- 1$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - 2 (- \frac{1}{2 x^{2}} + C_{4}) - 3 \int x^{- 4} d x$

Step 4.3.10

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

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

Step 4.3.11

Simplify.

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Step 4.3.11.1

Simplify.

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Step 4.3.11.1.1

Combine $x^{- 3}$ and $\frac{1}{3}$ .

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

Step 4.3.11.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.3.11.2

Simplify.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - \frac{- 1}{x^{2}} - 3 (- \frac{1}{3 x^{3}}) + C_{6}$

Step 4.3.11.3

Simplify.

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Step 4.3.11.3.1

Move the negative in front of the fraction.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = - - \frac{1}{x^{2}} - 3 (- \frac{1}{3 x^{3}}) + C_{6}$

Step 4.3.11.3.2

Multiply $- 1$ by $- 1$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = 1 \frac{1}{x^{2}} - 3 (- \frac{1}{3 x^{3}}) + C_{6}$

Step 4.3.11.3.3

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

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = \frac{1}{x^{2}} - 3 (- \frac{1}{3 x^{3}}) + C_{6}$

Step 4.3.11.3.4

Multiply $- 1$ by $- 3$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = \frac{1}{x^{2}} + 3 \frac{1}{3 x^{3}} + C_{6}$

Step 4.3.11.3.5

Combine $3$ and $\frac{1}{3 x^{3}}$ .

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = \frac{1}{x^{2}} + \frac{3}{3 x^{3}} + C_{6}$

Step 4.3.11.3.6

Cancel the common factor of $3$ .

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Step 4.3.11.3.6.1

Cancel the common factor.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = \frac{1}{x^{2}} + \frac{3}{3 x^{3}} + C_{6}$

Step 4.3.11.3.6.2

Rewrite the expression.

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} + C_{3} = \frac{1}{x^{2}} + \frac{1}{x^{3}} + C_{6}$

Step 4.4

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

$- \frac{1}{y^{4}} - \frac{1}{y^{5}} = \frac{1}{x^{2}} + \frac{1}{x^{3}} + K$