Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

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

Step 2.2.1.2

Move $y^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2.1.3

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.3.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 2.2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = x + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = x + K$ by $2$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Divide $y^{\frac{1}{2}}$ by $1$ .

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

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify ${(\frac{x}{2} + \frac{K}{2})}^{2}$ .

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Step 3.3.2.1.1

Rewrite ${(\frac{x}{2} + \frac{K}{2})}^{2}$ as $(\frac{x}{2} + \frac{K}{2}) (\frac{x}{2} + \frac{K}{2})$ .

$y = (\frac{x}{2} + \frac{K}{2}) (\frac{x}{2} + \frac{K}{2})$

Step 3.3.2.1.2

Expand $(\frac{x}{2} + \frac{K}{2}) (\frac{x}{2} + \frac{K}{2})$ using the FOIL Method.

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Step 3.3.2.1.2.1

Apply the distributive property.

$y = \frac{x}{2} (\frac{x}{2} + \frac{K}{2}) + \frac{K}{2} (\frac{x}{2} + \frac{K}{2})$

Step 3.3.2.1.2.2

Apply the distributive property.

$y = \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} (\frac{x}{2} + \frac{K}{2})$

Step 3.3.2.1.2.3

Apply the distributive property.

$y = \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3

Simplify and combine like terms.

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Step 3.3.2.1.3.1

Simplify each term.

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Step 3.3.2.1.3.1.1

Multiply $\frac{x}{2} \cdot \frac{x}{2}$ .

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Step 3.3.2.1.3.1.1.1

Multiply $\frac{x}{2}$ by $\frac{x}{2}$ .

$y = \frac{x \cdot x}{2 \cdot 2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.1.2

Raise $x$ to the power of $1$ .

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

Step 3.3.2.1.3.1.1.3

Raise $x$ to the power of $1$ .

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

Step 3.3.2.1.3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.2.1.3.1.1.5

Add $1$ and $1$ .

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

Step 3.3.2.1.3.1.1.6

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2

Multiply $\frac{x}{2} \cdot \frac{K}{2}$ .

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Step 3.3.2.1.3.1.2.1

Multiply $\frac{x}{2}$ by $\frac{K}{2}$ .

$y = \frac{x^{2}}{4} + \frac{x K}{2 \cdot 2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.3

Multiply $\frac{K}{2} \cdot \frac{x}{2}$ .

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Step 3.3.2.1.3.1.3.1

Multiply $\frac{K}{2}$ by $\frac{x}{2}$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{2 \cdot 2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.3.2

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4

Multiply $\frac{K}{2} \cdot \frac{K}{2}$ .

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Step 3.3.2.1.3.1.4.1

Multiply $\frac{K}{2}$ by $\frac{K}{2}$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} + \frac{K \cdot K}{2 \cdot 2}$

Step 3.3.2.1.3.1.4.2

Raise $K$ to the power of $1$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} + \frac{K^{1} K}{2 \cdot 2}$

Step 3.3.2.1.3.1.4.3

Raise $K$ to the power of $1$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} + \frac{K^{1} K^{1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} + \frac{K^{1 + 1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.4.5

Add $1$ and $1$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} + \frac{K^{2}}{2 \cdot 2}$

Step 3.3.2.1.3.1.4.6

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2

Add $\frac{x K}{4}$ and $\frac{K x}{4}$ .

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Step 3.3.2.1.3.2.1

Reorder $x$ and $K$ .

$y = \frac{x^{2}}{4} + \frac{K x}{4} + \frac{K x}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2.2

Add $\frac{K x}{4}$ and $\frac{K x}{4}$ .

$y = \frac{x^{2}}{4} + 2 \frac{K x}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.1

Factor $2$ out of $4$ .

$y = \frac{x^{2}}{4} + 2 \frac{K x}{2 (2)} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.2

Cancel the common factor.

$y = \frac{x^{2}}{4} + 2 \frac{K x}{2 \cdot 2} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.3

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{K x}{2} + \frac{K^{2}}{4}$

Step 3.4

Simplify $\frac{x^{2}}{4} + \frac{K x}{2} + \frac{K^{2}}{4}$ .

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Step 3.4.1

Move $\frac{x^{2}}{4}$ .

$y = \frac{K x}{2} + \frac{K^{2}}{4} + \frac{x^{2}}{4}$

Step 3.4.2

Reorder $\frac{K x}{2}$ and $\frac{K^{2}}{4}$ .

$y = \frac{K^{2}}{4} + \frac{K x}{2} + \frac{x^{2}}{4}$

Step 4

Simplify the constant of integration.

$y = K + \frac{K x}{2} + \frac{x^{2}}{4}$