Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

Step 2.3

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

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

Step 2.4

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

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