Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

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

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

Step 2.3.4

Simplify.

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

Step 2.3.5

Reorder terms.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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