Calculus Examples

$\frac{d y}{d x} + y = \frac{1}{1 + e^{x}}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 1$ .

Tap for more steps...

Step 1.1

Set up the integration.

$e^{\int d x}$

Step 1.2

Apply the constant rule.

$e^{x + C}$

Step 1.3

Remove the constant of integration.

$e^{x}$

Step 2

Multiply each term by the integrating factor $e^{x}$ .

Tap for more steps...

Step 2.1

Multiply each term by $e^{x}$ .

$e^{x} \frac{d y}{d x} + e^{x} y = e^{x} \frac{1}{1 + e^{x}}$

Step 2.2

Combine $e^{x}$ and $\frac{1}{1 + e^{x}}$ .

$e^{x} \frac{d y}{d x} + e^{x} y = \frac{e^{x}}{1 + e^{x}}$

Step 2.3

Reorder factors in $e^{x} \frac{d y}{d x} + e^{x} y = \frac{e^{x}}{1 + e^{x}}$ .

$e^{x} \frac{d y}{d x} + y e^{x} = \frac{e^{x}}{1 + e^{x}}$

Step 3

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

$\frac{d}{d x} [e^{x} y] = \frac{e^{x}}{1 + e^{x}}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x} y] d x = \int \frac{e^{x}}{1 + e^{x}} d x$

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

Tap for more steps...

Step 6.1

Let $u = 1 + e^{x}$ . Then $d u = e^{x} d x$ , so $\frac{1}{e^{x}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 6.1.1

Let $u = 1 + e^{x}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 6.1.1.1

Differentiate $1 + e^{x}$ .

$\frac{d}{d x} [1 + e^{x}]$

Step 6.1.1.2

Differentiate.

Tap for more steps...

Step 6.1.1.2.1

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

$\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}]$

Step 6.1.1.2.2

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

$0 + \frac{d}{d x} [e^{x}]$

Step 6.1.1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$0 + e^{x}$

Step 6.1.1.4

Add $0$ and $e^{x}$ .

$e^{x}$

Step 6.1.2

Rewrite the problem using $u$ and $d u$ .

$e^{x} y = \int \frac{1}{u} d u$

Step 6.2

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

$e^{x} y = \ln (| u |) + C$

Step 6.3

Replace all occurrences of $u$ with $1 + e^{x}$ .

$e^{x} y = \ln (| 1 + e^{x} |) + C$

Step 7

Divide each term in $e^{x} y = \ln (| 1 + e^{x} |) + C$ by $e^{x}$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $e^{x} y = \ln (| 1 + e^{x} |) + C$ by $e^{x}$ .

$\frac{e^{x} y}{e^{x}} = \frac{\ln (| 1 + e^{x} |)}{e^{x}} + \frac{C}{e^{x}}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

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

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{e^{x} y}{e^{x}} = \frac{\ln (| 1 + e^{x} |)}{e^{x}} + \frac{C}{e^{x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (| 1 + e^{x} |)}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Combine the numerators over the common denominator.

$y = \frac{\ln (| 1 + e^{x} |) + C}{e^{x}}$