Calculus Examples

$(1 + \ln (x)) d x + (1 + \ln (y)) d y = 0$

Step 1

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

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int 1 + \ln (y) d y = \int - (1 + \ln (x)) d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Split the single integral into multiple integrals.

$\int d y + \int \ln (y) d y = \int - (1 + \ln (x)) d x$

Step 2.2.2

Apply the constant rule.

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

Step 2.2.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (y)$ and $d v = 1$ .

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

Step 2.2.4

Simplify.

Tap for more steps...

Step 2.2.4.1

Combine $y$ and $\frac{1}{y}$ .

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

Step 2.2.4.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 2.2.4.2.1

Cancel the common factor.

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

Step 2.2.4.2.2

Rewrite the expression.

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

Step 2.2.5

Apply the constant rule.

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

Step 2.2.6

Simplify.

Tap for more steps...

Step 2.2.6.1

Simplify.

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

Step 2.2.6.2

Simplify.

Tap for more steps...

Step 2.2.6.2.1

Subtract $y$ from $y$ .

$\ln (y) y + 0 + C_{3} = \int - (1 + \ln (x)) d x$

Step 2.2.6.2.2

Add $\ln (y) y$ and $0$ .

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

Step 2.2.7

Reorder terms.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

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

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

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

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

Step 2.3.5.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.5.2.1

Cancel the common factor.

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

Step 2.3.5.2.2

Rewrite the expression.

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

Step 2.3.6

Apply the constant rule.

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

Step 2.3.7

Simplify.

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

Step 2.3.8

Reorder terms.

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

Step 2.4

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

$y \ln (y) = - x \ln (x) + K$