Calculus Examples

$(1 + x) (y d) x + (1 - y) x d y = 0$

Step 1

Subtract $(1 + x) y d x$ from both sides of the equation.

$(1 - y) x d y = - (1 + x) y d x$

Step 2

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.1.1

Factor $x$ out of $x y$ .

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

Step 3.1.2

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

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

Step 3.1.3

Cancel the common factor.

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

Step 3.1.4

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Factor $y$ out of $x y$ .

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

Step 3.4.3

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

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

Step 3.4.4

Cancel the common factor.

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

Step 3.4.5

Rewrite the expression.

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

Step 3.5

Move the negative in front of the fraction.

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

Step 3.6

Apply the distributive property.

$\frac{1 - y}{y} d y = (- \frac{1}{x} \cdot 1 - \frac{1}{x} x) d x$

Step 3.7

Multiply $- 1$ by $1$ .

$\frac{1 - y}{y} d y = (- \frac{1}{x} - \frac{1}{x} x) d x$

Step 3.8

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.8.1

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

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

Step 3.8.2

Cancel the common factor.

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

Step 3.8.3

Rewrite the expression.

$\frac{1 - y}{y} d y = (- \frac{1}{x} - 1) d x$

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

$\int \frac{1 - y}{y} d y = \int - \frac{1}{x} - 1 d x$

Step 4.2

Integrate the left side.

Tap for more steps...

Step 4.2.1

Split the fraction into multiple fractions.

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

Step 4.2.2

Split the single integral into multiple integrals.

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

Step 4.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Divide $- 1$ by $1$ .

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

Step 4.2.4

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

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

Step 4.2.5

Apply the constant rule.

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

Step 4.2.6

Simplify.

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

Step 4.2.7

Reorder terms.

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

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

Split the single integral into multiple integrals.

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Apply the constant rule.

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

Step 4.3.5

Simplify.

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

Step 4.3.6

Reorder terms.

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

Step 4.4

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

$- y + \ln (| y |) = - x - \ln (| x |) + C$