Calculus Examples

$(1 + x^{2}) \frac{d y}{d x} + y = \arctan (x)$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

Tap for more steps...

Step 1.1

Reorder terms.

$(x^{2} + 1) \frac{d y}{d x} + y = \arctan (x)$

Step 1.2

Divide each term in $(x^{2} + 1) \frac{d y}{d x} + y = \arctan (x)$ by $x^{2} + 1$ .

$\frac{(x^{2} + 1) \frac{d y}{d x}}{x^{2} + 1} + \frac{y}{x^{2} + 1} = \frac{\arctan (x)}{x^{2} + 1}$

Step 1.3

Cancel the common factor of $x^{2} + 1$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$\frac{(x^{2} + 1) \frac{d y}{d x}}{x^{2} + 1} + \frac{y}{x^{2} + 1} = \frac{\arctan (x)}{x^{2} + 1}$

Step 1.3.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} + \frac{y}{x^{2} + 1} = \frac{\arctan (x)}{x^{2} + 1}$

Step 1.4

Factor $y$ out of $\frac{y}{x^{2} + 1}$ .

$\frac{d y}{d x} + y \frac{1}{x^{2} + 1} = \frac{\arctan (x)}{x^{2} + 1}$

Step 1.5

Reorder $y$ and $\frac{1}{x^{2} + 1}$ .

$\frac{d y}{d x} + \frac{1}{x^{2} + 1} y = \frac{\arctan (x)}{x^{2} + 1}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x^{2} + 1}$ .

Tap for more steps...

Step 2.1

Set up the integration.

$e^{\int \frac{1}{x^{2} + 1} d x}$

Step 2.2

Integrate $\frac{1}{x^{2} + 1}$ .

Tap for more steps...

Step 2.2.1

Simplify the expression.

Tap for more steps...

Step 2.2.1.1

Reorder $x^{2}$ and $1$ .

$e^{\int \frac{1}{1 + x^{2}} d x}$

Step 2.2.1.2

Rewrite $1$ as $1^{2}$ .

$e^{\int \frac{1}{1^{2} + x^{2}} d x}$

Step 2.2.2

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C$ .

$e^{\arctan (x) + C}$

Step 2.3

Remove the constant of integration.

$e^{\arctan (x)}$

Step 3

Multiply each term by the integrating factor $e^{\arctan (x)}$ .

Tap for more steps...

Step 3.1

Multiply each term by $e^{\arctan (x)}$ .

$e^{\arctan (x)} \frac{d y}{d x} + e^{\arctan (x)} (\frac{1}{x^{2} + 1} y) = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

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

$e^{\arctan (x)} \frac{d y}{d x} + e^{\arctan (x)} \frac{y}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.2.2

Combine $e^{\arctan (x)}$ and $\frac{y}{x^{2} + 1}$ .

$e^{\arctan (x)} \frac{d y}{d x} + \frac{e^{\arctan (x)} y}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.3

To write $e^{\arctan (x)} \frac{d y}{d x}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 1}{x^{2} + 1}$ .

$\frac{e^{\arctan (x)} \frac{d y}{d x} (x^{2} + 1)}{x^{2} + 1} + \frac{e^{\arctan (x)} y}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{e^{\arctan (x)} \frac{d y}{d x} (x^{2} + 1) + e^{\arctan (x)} y}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Factor $e^{\arctan (x)}$ out of $e^{\arctan (x)} \frac{d y}{d x} (x^{2} + 1) + e^{\arctan (x)} y$ .

Tap for more steps...

Step 3.5.1.1

Factor $e^{\arctan (x)}$ out of $e^{\arctan (x)} \frac{d y}{d x} (x^{2} + 1)$ .

$\frac{e^{\arctan (x)} (\frac{d y}{d x} (x^{2} + 1)) + e^{\arctan (x)} y}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.5.1.2

Factor $e^{\arctan (x)}$ out of $e^{\arctan (x)} y$ .

$\frac{e^{\arctan (x)} (\frac{d y}{d x} (x^{2} + 1)) + e^{\arctan (x)} (y)}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.5.1.3

Factor $e^{\arctan (x)}$ out of $e^{\arctan (x)} (\frac{d y}{d x} (x^{2} + 1)) + e^{\arctan (x)} (y)$ .

$\frac{e^{\arctan (x)} (\frac{d y}{d x} (x^{2} + 1) + y)}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.5.2

Apply the distributive property.

$\frac{e^{\arctan (x)} (\frac{d y}{d x} x^{2} + \frac{d y}{d x} \cdot 1 + y)}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.5.3

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

$\frac{e^{\arctan (x)} (\frac{d y}{d x} x^{2} + \frac{d y}{d x} + y)}{x^{2} + 1} = e^{\arctan (x)} \frac{\arctan (x)}{x^{2} + 1}$

Step 3.6

Combine $e^{\arctan (x)}$ and $\frac{\arctan (x)}{x^{2} + 1}$ .

$\frac{e^{\arctan (x)} (\frac{d y}{d x} x^{2} + \frac{d y}{d x} + y)}{x^{2} + 1} = \frac{e^{\arctan (x)} \arctan (x)}{x^{2} + 1}$

Step 3.7

Reorder factors in $\frac{e^{\arctan (x)} (\frac{d y}{d x} x^{2} + \frac{d y}{d x} + y)}{x^{2} + 1} = \frac{e^{\arctan (x)} \arctan (x)}{x^{2} + 1}$ .

$\frac{e^{\arctan (x)} (x^{2} \frac{d y}{d x} + \frac{d y}{d x} + y)}{x^{2} + 1} = \frac{e^{\arctan (x)} \arctan (x)}{x^{2} + 1}$

Step 4

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

$\frac{d}{d x} [e^{\arctan (x)} y] = \frac{e^{\arctan (x)} \arctan (x)}{x^{2} + 1}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{\arctan (x)} y] d x = \int \frac{e^{\arctan (x)} \arctan (x)}{x^{2} + 1} d x$

Step 6

Integrate the left side.

$e^{\arctan (x)} y = \int \frac{e^{\arctan (x)} \arctan (x)}{x^{2} + 1} d x$

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

Let $u = \arctan (x)$ . Then $d u = \frac{1}{x^{2} + 1} d x$ , so $(x^{2} + 1) d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 7.1.1

Let $u = \arctan (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 7.1.1.1

Differentiate $\arctan (x)$ .

$\frac{d}{d x} [\arctan (x)]$

Step 7.1.1.2

The derivative of $\arctan (x)$ with respect to $x$ is $\frac{1}{1 + x^{2}}$ .

$\frac{1}{1 + x^{2}}$

Step 7.1.1.3

Reorder terms.

$\frac{1}{x^{2} + 1}$

Step 7.1.2

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

$e^{\arctan (x)} y = \int e^{u} u d u$

Step 7.2

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = e^{u}$ .

$e^{\arctan (x)} y = u e^{u} - \int e^{u} d u$

Step 7.3

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{\arctan (x)} y = u e^{u} - (e^{u} + C)$

Step 7.4

Simplify.

$e^{\arctan (x)} y = u e^{u} - e^{u} + C$

Step 7.5

Replace all occurrences of $u$ with $\arctan (x)$ .

$e^{\arctan (x)} y = \arctan (x) e^{\arctan (x)} - e^{\arctan (x)} + C$

Step 8

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

Tap for more steps...

Step 8.1

Divide each term in $e^{\arctan (x)} y = \arctan (x) e^{\arctan (x)} - e^{\arctan (x)} + C$ by $e^{\arctan (x)}$ .

$\frac{e^{\arctan (x)} y}{e^{\arctan (x)}} = \frac{\arctan (x) e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{- e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{C}{e^{\arctan (x)}}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $e^{\arctan (x)}$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{e^{\arctan (x)} y}{e^{\arctan (x)}} = \frac{\arctan (x) e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{- e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{C}{e^{\arctan (x)}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arctan (x) e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{- e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{C}{e^{\arctan (x)}}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Simplify each term.

Tap for more steps...

Step 8.3.1.1

Cancel the common factor of $e^{\arctan (x)}$ .

Tap for more steps...

Step 8.3.1.1.1

Cancel the common factor.

$y = \frac{\arctan (x) e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{- e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{C}{e^{\arctan (x)}}$

Step 8.3.1.1.2

Divide $\arctan (x)$ by $1$ .

$y = \arctan (x) + \frac{- e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{C}{e^{\arctan (x)}}$

Step 8.3.1.2

Cancel the common factor of $e^{\arctan (x)}$ .

Tap for more steps...

Step 8.3.1.2.1

Cancel the common factor.

$y = \arctan (x) + \frac{- e^{\arctan (x)}}{e^{\arctan (x)}} + \frac{C}{e^{\arctan (x)}}$

Step 8.3.1.2.2

Divide $- 1$ by $1$ .

$y = \arctan (x) - 1 + \frac{C}{e^{\arctan (x)}}$