Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Factor.

Tap for more steps...

Step 1.1.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.1.1.1

Group the first two terms and the last two terms.

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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

Step 1.2

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

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

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

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

Step 2.3.4

Simplify.

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

Step 2.3.5

Reorder terms.

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

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 $y$ .

Tap for more steps...

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.

Tap for more steps...

Step 3.2.1

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

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