Calculus Examples

$(x - 4) y^{4} d x - x^{3} (y^{2} - 3) d y = 0$

Step 1

Subtract $(x - 4) y^{4} d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $\frac{1}{x^{3} y^{4}}$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Cancel the common factor of $x^{3}$ .

Tap for more steps...

Step 3.2.1

Move the leading negative in $- \frac{1}{x^{3} y^{4}}$ into the numerator.

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

Step 3.2.2

Factor $x^{3}$ out of $x^{3} y^{4}$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Move the negative in front of the fraction.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

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

Step 3.6

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

Tap for more steps...

Step 3.6.1

Multiply $- 3$ by $- 1$ .

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

Step 3.6.2

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

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

Step 3.7

Move the negative in front of the fraction.

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

Step 3.8

Rewrite using the commutative property of multiplication.

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

Step 3.9

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

Tap for more steps...

Step 3.9.1

Move the leading negative in $- \frac{1}{x^{3} y^{4}}$ into the numerator.

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

Step 3.9.2

Factor $y^{4}$ out of $x^{3} y^{4}$ .

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

Step 3.9.3

Factor $y^{4}$ out of $(x - 4) y^{4}$ .

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

Step 3.9.4

Cancel the common factor.

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

Step 3.9.5

Rewrite the expression.

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

Step 3.10

Move the negative in front of the fraction.

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

Step 3.11

Apply the distributive property.

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

Step 3.12

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.12.1

Move the leading negative in $- \frac{1}{x^{3}}$ into the numerator.

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

Step 3.12.2

Factor $x$ out of $x^{3}$ .

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

Step 3.12.3

Cancel the common factor.

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

Step 3.12.4

Rewrite the expression.

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

Step 3.13

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

Tap for more steps...

Step 3.13.1

Multiply $- 4$ by $- 1$ .

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

Step 3.13.2

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

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

Step 3.14

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

Tap for more steps...

Step 4.2.1

Split the single integral into multiple integrals.

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

Step 4.2.2

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

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

Step 4.2.3

Apply basic rules of exponents.

Tap for more steps...

Step 4.2.3.1

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

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

Step 4.2.3.2

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

Tap for more steps...

Step 4.2.3.2.1

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

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

Step 4.2.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.2.4

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

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

Step 4.2.5

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

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

Step 4.2.6

Apply basic rules of exponents.

Tap for more steps...

Step 4.2.6.1

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

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

Step 4.2.6.2

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

Tap for more steps...

Step 4.2.6.2.1

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

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

Step 4.2.6.2.2

Multiply $4$ by $- 1$ .

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

Step 4.2.7

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

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

Step 4.2.8

Simplify.

Tap for more steps...

Step 4.2.8.1

Simplify.

Tap for more steps...

Step 4.2.8.1.1

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

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

Step 4.2.8.1.2

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

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

Step 4.2.8.2

Simplify.

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

Step 4.2.8.3

Simplify.

Tap for more steps...

Step 4.2.8.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.8.3.2

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

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

Step 4.2.8.3.3

Multiply $- 1$ by $3$ .

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

Step 4.2.8.3.4

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

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

Step 4.2.8.3.5

Cancel the common factor of $- 3$ and $3$ .

Tap for more steps...

Step 4.2.8.3.5.1

Factor $3$ out of $- 3$ .

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

Step 4.2.8.3.5.2

Cancel the common factors.

Tap for more steps...

Step 4.2.8.3.5.2.1

Factor $3$ out of $3 y^{3}$ .

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

Step 4.2.8.3.5.2.2

Cancel the common factor.

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

Step 4.2.8.3.5.2.3

Rewrite the expression.

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

Step 4.2.8.3.6

Move the negative in front of the fraction.

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

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

Split the single integral into multiple integrals.

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

Step 4.3.2

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

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

Step 4.3.3

Apply basic rules of exponents.

Tap for more steps...

Step 4.3.3.1

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

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

Step 4.3.3.2

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

Tap for more steps...

Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Apply basic rules of exponents.

Tap for more steps...

Step 4.3.6.1

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

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

Step 4.3.6.2

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

Tap for more steps...

Step 4.3.6.2.1

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

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

Step 4.3.6.2.2

Multiply $3$ by $- 1$ .

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

Step 4.3.7

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

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

Step 4.3.8

Simplify.

Tap for more steps...

Step 4.3.8.1

Simplify.

Tap for more steps...

Step 4.3.8.1.1

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

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

Step 4.3.8.1.2

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

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

Step 4.3.8.2

Simplify.

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

Step 4.3.8.3

Simplify.

Tap for more steps...

Step 4.3.8.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.8.3.2

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

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

Step 4.3.8.3.3

Multiply $- 1$ by $4$ .

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

Step 4.3.8.3.4

Combine $- 4$ and $\frac{1}{2 x^{2}}$ .

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

Step 4.3.8.3.5

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

Tap for more steps...

Step 4.3.8.3.5.1

Factor $2$ out of $- 4$ .

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

Step 4.3.8.3.5.2

Cancel the common factors.

Tap for more steps...

Step 4.3.8.3.5.2.1

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

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

Step 4.3.8.3.5.2.2

Cancel the common factor.

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

Step 4.3.8.3.5.2.3

Rewrite the expression.

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

Step 4.3.8.3.6

Move the negative in front of the fraction.

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

Step 4.4

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

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