Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

Apply the constant rule.

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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

Step 2.2.5

Reorder terms.

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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