Calculus Examples

$\frac{d y}{d x} = \frac{1 + \sqrt{x}}{1 + \sqrt{y}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $1 + \sqrt{y}$ .

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

Step 1.2

Cancel the common factor of $1 + \sqrt{y}$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$(1 + \sqrt{y}) d y = (1 + \sqrt{x}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 1 + \sqrt{y} d y = \int 1 + \sqrt{x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int d y + \int \sqrt{y} d y = \int 1 + \sqrt{x} d x$

Step 2.2.2

Apply the constant rule.

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

Step 2.2.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.2.6

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

$\frac{2}{3} y^{\frac{3}{2}} + y + C_{3} = x + C_{4} + \int \sqrt{x} d x$

Step 2.3.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{2}{3} y^{\frac{3}{2}} + y + C_{3} = x + C_{4} + \int x^{\frac{1}{2}} d x$

Step 2.3.4

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

$\frac{2}{3} y^{\frac{3}{2}} + y + C_{3} = x + C_{4} + \frac{2}{3} x^{\frac{3}{2}} + C_{5}$

Step 2.3.5

Simplify.

$\frac{2}{3} y^{\frac{3}{2}} + y + C_{3} = x + \frac{2}{3} x^{\frac{3}{2}} + C_{6}$

Step 2.3.6

Reorder terms.

$\frac{2}{3} y^{\frac{3}{2}} + y + C_{3} = \frac{2}{3} x^{\frac{3}{2}} + x + C_{6}$

Step 2.4

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

$\frac{2}{3} y^{\frac{3}{2}} + y = \frac{2}{3} x^{\frac{3}{2}} + x + K$