Calculus Examples

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

Step 1

Check if the left side of the equation is the result of the derivative of the term $(x + 1) y$ .

Tap for more steps...

Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 1$ and $g (x) = y$ .

$(x + 1) \frac{d}{d x} [y] + y \frac{d}{d x} [x + 1]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$(x + 1) y' + y \frac{d}{d x} [x + 1]$

Step 1.3

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$(x + 1) y' + y (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$(x + 1) y' + y (1 + \frac{d}{d x} [1])$

Step 1.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(x + 1) y' + y (1 + 0)$

Step 1.6

Add $1$ and $0$ .

$(x + 1) y' + y \cdot 1$

Step 1.7

Substitute $\frac{d y}{d x}$ for $y'$ .

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

Step 1.8

Reorder $y$ and $1$ .

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

Step 1.9

Multiply $y$ by $1$ .

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

Step 2

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [(x + 1) y] = \ln (x)$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [(x + 1) y] d x = \int \ln (x) d x$

Step 4

Integrate the left side.

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

Step 5

Integrate the right side.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Rewrite the expression.

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

Step 5.3

Apply the constant rule.

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

Step 5.4

Simplify.

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

Step 6

Divide each term in $(x + 1) y = \ln (x) x - x + C$ by $x + 1$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $(x + 1) y = \ln (x) x - x + C$ by $x + 1$ .

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

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $y$ by $1$ .

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

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Move the negative in front of the fraction.

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

Step 6.3.2

Combine the numerators over the common denominator.

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Reorder factors in $\ln (x) x - x + C$ .

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