Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Combine the numerators over the common denominator.

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify the expression.

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.2

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

$\arctan (y) + C_{1} = \arctan (x) + C_{2}$

Step 2.4

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

$\arctan (y) = \arctan (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\arctan (x) + K = \arctan (y)$ .

$\arctan (x) + K = \arctan (y)$

Step 3.2

Subtract $K$ from both sides of the equation.

$\arctan (x) = \arctan (y) - K$

Step 3.3

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$x = \tan (\arctan (y) - K)$

Step 3.4

Rewrite the equation as $\tan (\arctan (y) - K) = x$ .

$\tan (\arctan (y) - K) = x$

Step 3.5

Take the inverse tangent of both sides of the equation to extract $\arctan (y)$ from inside the tangent.

$\arctan (y) - K = \arctan (x)$

Step 3.6

Add $K$ to both sides of the equation.

$\arctan (y) = \arctan (x) + K$

Step 3.7

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$y = \tan (\arctan (x) + K)$

Step 3.8

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\tan (\arctan (y) - K) = x$

Step 3.9

Take the inverse tangent of both sides of the equation to extract $\arctan (y)$ from inside the tangent.

$\arctan (y) - K = \arctan (x)$

Step 3.10

Add $K$ to both sides of the equation.

$\arctan (y) = \arctan (x) + K$

Step 3.11

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$y = \tan (\arctan (x) + K)$