Calculus Examples

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

Step 1

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

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Step 1.1

Factor $y$ out of $\frac{y}{x}$ .

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

Step 1.2

Reorder $y$ and $\frac{1}{x}$ .

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 2.4

Exponentiation and log are inverse functions.

$x$

Step 3

Multiply each term by the integrating factor $x$ .

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Step 3.1

Multiply each term by $x$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

$x y = \int x \arctan (x) d x$

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.4

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

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

Step 7.5

Divide $x^{2}$ by $x^{2} + 1$ .

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Step 7.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$0$

Step 7.5.2

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x^{2}$ .

							$1$
	$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$

Step 7.5.3

Multiply the new quotient term by the divisor.

						$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
					+	$x^{2}$	+	$0$	+	$1$

Step 7.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0 + 1$

						$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
					-	$x^{2}$	-	$0$	-	$1$

Step 7.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

						$1$
$x^{2}$	+	$0 x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
					-	$x^{2}$	-	$0$	-	$1$
									-	$1$

Step 7.5.6

The final answer is the quotient plus the remainder over the divisor.

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

Step 7.6

Split the single integral into multiple integrals.

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

Step 7.7

Apply the constant rule.

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

Step 7.8

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

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

Step 7.9

Simplify the expression.

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Step 7.9.1

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

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

Step 7.9.2

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

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

Step 7.10

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

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

Step 7.11

Simplify.

$x y = \frac{\arctan (x) x^{2}}{2} - \frac{x}{2} + \frac{\arctan (x)}{2} + C$

Step 7.12

Reorder terms.

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

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

Combine $x$ and $\frac{1}{2}$ .

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

Step 8.1.2

Remove parentheses.

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

Step 8.2

Divide each term in $x y = \frac{1}{2} \cdot (\arctan (x) x^{2}) - \frac{x}{2} + \frac{1}{2} \cdot \arctan (x) + C$ by $x$ and simplify.

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $x$ .

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

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

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Step 8.2.3.1.1.1

Factor $x$ out of $\frac{1}{2} \cdot (\arctan (x) x^{2})$ .

$y = \frac{x (\frac{1}{2} \cdot (\arctan (x) x))}{x} + \frac{- \frac{x}{2}}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.1.2

Cancel the common factors.

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Step 8.2.3.1.1.2.1

Raise $x$ to the power of $1$ .

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

Step 8.2.3.1.1.2.2

Factor $x$ out of $x^{1}$ .

$y = \frac{x (\frac{1}{2} \cdot (\arctan (x) x))}{x \cdot 1} + \frac{- \frac{x}{2}}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.1.2.3

Cancel the common factor.

$y = \frac{x (\frac{1}{2} \cdot (\arctan (x) x))}{x \cdot 1} + \frac{- \frac{x}{2}}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.1.2.4

Rewrite the expression.

$y = \frac{\frac{1}{2} \cdot (\arctan (x) x)}{1} + \frac{- \frac{x}{2}}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.1.2.5

Divide $\frac{1}{2} \cdot (\arctan (x) x)$ by $1$ .

$y = \frac{1}{2} \cdot (\arctan (x) x) + \frac{- \frac{x}{2}}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.2

Multiply $\frac{1}{2} (\arctan (x) x)$ .

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Step 8.2.3.1.2.1

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

$y = \frac{\arctan (x)}{2} x + \frac{- \frac{x}{2}}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.2.2

Combine $\frac{\arctan (x)}{2}$ and $x$ .

$y = \frac{\arctan (x) x}{2} + \frac{- \frac{x}{2}}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{\arctan (x) x}{2} - \frac{x}{2} \cdot \frac{1}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.4

Cancel the common factor of $x$ .

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Step 8.2.3.1.4.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$y = \frac{\arctan (x) x}{2} + \frac{- x}{2} \cdot \frac{1}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.4.2

Factor $x$ out of $- x$ .

$y = \frac{\arctan (x) x}{2} + \frac{x \cdot - 1}{2} \cdot \frac{1}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.4.3

Cancel the common factor.

$y = \frac{\arctan (x) x}{2} + \frac{x \cdot - 1}{2} \cdot \frac{1}{x} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.4.4

Rewrite the expression.

$y = \frac{\arctan (x) x}{2} + \frac{- 1}{2} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.5

Move the negative in front of the fraction.

$y = \frac{\arctan (x) x}{2} - \frac{1}{2} + \frac{\frac{1}{2} \cdot \arctan (x)}{x} + \frac{C}{x}$

Step 8.2.3.1.6

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

$y = \frac{\arctan (x) x}{2} - \frac{1}{2} + \frac{\frac{\arctan (x)}{2}}{x} + \frac{C}{x}$

Step 8.2.3.1.7

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{\arctan (x) x}{2} - \frac{1}{2} + \frac{\arctan (x)}{2} \cdot \frac{1}{x} + \frac{C}{x}$

Step 8.2.3.1.8

Multiply $\frac{\arctan (x)}{2}$ by $\frac{1}{x}$ .

$y = \frac{\arctan (x) x}{2} - \frac{1}{2} + \frac{\arctan (x)}{2 x} + \frac{C}{x}$

Step 8.2.3.2

Reorder factors in $\frac{\arctan (x) x}{2} - \frac{1}{2} + \frac{\arctan (x)}{2 x} + \frac{C}{x}$ .

$y = \frac{x \arctan (x)}{2} - \frac{1}{2} + \frac{\arctan (x)}{2 x} + \frac{C}{x}$