Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{x^{2}}{1 + x^{2}} \cdot \frac{1}{y}$

Step 1.2

Multiply both sides by $y$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $\frac{x^{2}}{1 + x^{2}}$ by $\frac{1}{y}$ .

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

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

Factor $y$ out of $(1 + x^{2}) y$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x^{2}}{1 + x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x^{2}}{x^{2} + 1} d x$

Step 2.3.2

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

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Step 2.3.2.1

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 2.3.2.2

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 2.3.2.3

Multiply the new quotient term by the divisor.

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

Step 2.3.2.4

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 2.3.2.5

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 2.3.2.6

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

$\frac{1}{2} y^{2} + C_{1} = \int 1 - \frac{1}{x^{2} + 1} d x$

Step 2.3.3

Split the single integral into multiple integrals.

$\frac{1}{2} y^{2} + C_{1} = \int d x + \int - \frac{1}{x^{2} + 1} d x$

Step 2.3.4

Apply the constant rule.

$\frac{1}{2} y^{2} + C_{1} = x + C_{2} + \int - \frac{1}{x^{2} + 1} d x$

Step 2.3.5

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

$\frac{1}{2} y^{2} + C_{1} = x + C_{2} - \int \frac{1}{x^{2} + 1} d x$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

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

$\frac{1}{2} y^{2} + C_{1} = x + C_{2} - \int \frac{1}{1 + x^{2}} d x$

Step 2.3.6.2

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

$\frac{1}{2} y^{2} + C_{1} = x + C_{2} - \int \frac{1}{1^{2} + x^{2}} d x$

Step 2.3.7

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C_{3}$ .

$\frac{1}{2} y^{2} + C_{1} = x + C_{2} - (\arctan (x) + C_{3})$

Step 2.3.8

Simplify.

$\frac{1}{2} y^{2} + C_{1} = x - \arctan (x) + C_{4}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (x - \arctan (x) + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

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

Step 3.2.2.1.2

Multiply $- 1$ by $2$ .

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

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{2 x - 2 \arctan (x) + 2 K}$

Step 3.4

Factor $2$ out of $2 x - 2 \arctan (x) + 2 K$ .

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Step 3.4.1

Factor $2$ out of $2 x$ .

$y = \pm \sqrt{2 (x) - 2 \arctan (x) + 2 K}$

Step 3.4.2

Factor $2$ out of $- 2 \arctan (x)$ .

$y = \pm \sqrt{2 (x) + 2 (- \arctan (x)) + 2 K}$

Step 3.4.3

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (x) + 2 (- \arctan (x)) + 2 K}$

Step 3.4.4

Factor $2$ out of $2 (x) + 2 (- \arctan (x))$ .

$y = \pm \sqrt{2 (x - \arctan (x)) + 2 K}$

Step 3.4.5

Factor $2$ out of $2 (x - \arctan (x)) + 2 K$ .

$y = \pm \sqrt{2 (x - \arctan (x) + K)}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (x - \arctan (x) + K)}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (x - \arctan (x) + K)}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (x - \arctan (x) + K)}$

$y = - \sqrt{2 (x - \arctan (x) + K)}$

$y = \sqrt{2 (x - \arctan (x) + K)}$

$y = - \sqrt{2 (x - \arctan (x) + K)}$

$y = \sqrt{2 (x - \arctan (x) + K)}$

$y = - \sqrt{2 (x - \arctan (x) + K)}$