Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Factor.

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Step 1.1.1

Factor out the greatest common factor from each group.

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Step 1.1.1.1

Group the first two terms and the last two terms.

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

Step 1.1.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.1.2

Factor the polynomial by factoring out the greatest common factor, $x^{2} + 1$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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 $1$ .

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

Step 2.2.1.2

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

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

$\arctan (y) + C_{1} = \frac{1}{3} x^{3} + x + C_{4}$

Step 2.4

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

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

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.

$y = \tan (\frac{1}{3} x^{3} + x + K)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$y = \tan (\frac{x^{3}}{3} + x + K)$