Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 1 - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 1 - y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{- 1}{\sqrt{u}} d u = \int d x$

Step 2.2.2

Move the negative in front of the fraction.

$\int - \frac{1}{\sqrt{u}} d u = \int d x$

Step 2.2.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{1}{\sqrt{u}} d u = \int d x$

Step 2.2.4

Apply basic rules of exponents.

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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

Step 2.2.4.3

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

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Step 2.2.4.3.1

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

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

Step 2.2.4.3.2

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

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

Step 2.2.4.3.3

Move the negative in front of the fraction.

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

Step 2.2.5

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

$- (2 u^{\frac{1}{2}} + C_{1}) = \int d x$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Rewrite $- (2 u^{\frac{1}{2}} + C_{1})$ as $- 1 \cdot 2 u^{\frac{1}{2}} + C_{1}$ .

$- 1 \cdot 2 u^{\frac{1}{2}} + C_{1} = \int d x$

Step 2.2.6.2

Multiply $- 1$ by $2$ .

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

Step 2.2.7

Replace all occurrences of $u$ with $1 - y$ .

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative in front of the fraction.

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

Step 3.1.3.1.2

Move the negative in front of the fraction.

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

Step 3.2

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

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

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

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Step 3.3.1.1.1

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

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Step 3.3.1.1.1.1

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

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

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.

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

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

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

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 $(- \frac{x}{2} - \frac{K}{2}) (- \frac{x}{2} - \frac{K}{2})$ .

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

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.

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

Step 3.3.2.1.2.2

Apply the distributive property.

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

Step 3.3.2.1.2.3

Apply the distributive property.

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

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 $- \frac{x}{2} (- \frac{x}{2})$ .

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Step 3.3.2.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2.1.3.1.1.2

Multiply $\frac{x}{2}$ by $1$ .

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

Step 3.3.2.1.3.1.1.3

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

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

Step 3.3.2.1.3.1.1.4

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

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

Step 3.3.2.1.3.1.1.5

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

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

Step 3.3.2.1.3.1.1.6

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

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

Step 3.3.2.1.3.1.1.7

Add $1$ and $1$ .

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

Step 3.3.2.1.3.1.1.8

Multiply $2$ by $2$ .

$1 - y = \frac{x^{2}}{4} - \frac{x}{2} (- \frac{K}{2}) - \frac{K}{2} (- \frac{x}{2}) - \frac{K}{2} (- \frac{K}{2})$

Step 3.3.2.1.3.1.2

Multiply $- \frac{x}{2} (- \frac{K}{2})$ .

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Step 3.3.2.1.3.1.2.1

Multiply $- 1$ by $- 1$ .

$1 - y = \frac{x^{2}}{4} + 1 \frac{x}{2} \frac{K}{2} - \frac{K}{2} (- \frac{x}{2}) - \frac{K}{2} (- \frac{K}{2})$

Step 3.3.2.1.3.1.2.2

Multiply $\frac{x}{2}$ by $1$ .

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

Step 3.3.2.1.3.1.2.3

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

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

Step 3.3.2.1.3.1.2.4

Multiply $2$ by $2$ .

$1 - y = \frac{x^{2}}{4} + \frac{x K}{4} - \frac{K}{2} (- \frac{x}{2}) - \frac{K}{2} (- \frac{K}{2})$

Step 3.3.2.1.3.1.3

Multiply $- \frac{K}{2} (- \frac{x}{2})$ .

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Step 3.3.2.1.3.1.3.1

Multiply $- 1$ by $- 1$ .

$1 - y = \frac{x^{2}}{4} + \frac{x K}{4} + 1 \frac{K}{2} \frac{x}{2} - \frac{K}{2} (- \frac{K}{2})$

Step 3.3.2.1.3.1.3.2

Multiply $\frac{K}{2}$ by $1$ .

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

Step 3.3.2.1.3.1.3.3

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

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

Step 3.3.2.1.3.1.3.4

Multiply $2$ by $2$ .

$1 - y = \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x}{4} - \frac{K}{2} (- \frac{K}{2})$

Step 3.3.2.1.3.1.4

Multiply $- \frac{K}{2} (- \frac{K}{2})$ .

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Step 3.3.2.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2.1.3.1.4.2

Multiply $\frac{K}{2}$ by $1$ .

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

Step 3.3.2.1.3.1.4.3

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

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

Step 3.3.2.1.3.1.4.4

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

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

Step 3.3.2.1.3.1.4.5

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

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

Step 3.3.2.1.3.1.4.6

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

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

Step 3.3.2.1.3.1.4.7

Add $1$ and $1$ .

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

Step 3.3.2.1.3.1.4.8

Multiply $2$ by $2$ .

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

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 $K$ .

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

Step 3.3.2.1.3.2.2

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

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

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 $4$ .

$1 - y = \frac{x^{2}}{4} + 2 \frac{K x}{2 (2)} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.2

Cancel the common factor.

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

Step 3.3.2.1.4.3

Rewrite the expression.

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

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $1$ from both sides of the equation.

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

Step 3.4.2

Divide each term in $- y = \frac{x^{2}}{4} + \frac{K x}{2} + \frac{K^{2}}{4} - 1$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- y = \frac{x^{2}}{4} + \frac{K x}{2} + \frac{K^{2}}{4} - 1$ by $- 1$ .

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.4.2.2.2

Divide $y$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Move the negative one from the denominator of $\frac{\frac{x^{2}}{4}}{- 1}$ .

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

Step 3.4.2.3.1.2

Rewrite $- 1 \cdot \frac{x^{2}}{4}$ as $- \frac{x^{2}}{4}$ .

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

Step 3.4.2.3.1.3

Move the negative one from the denominator of $\frac{\frac{K x}{2}}{- 1}$ .

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

Step 3.4.2.3.1.4

Rewrite $- 1 \cdot \frac{K x}{2}$ as $- \frac{K x}{2}$ .

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

Step 3.4.2.3.1.5

Move the negative one from the denominator of $\frac{\frac{K^{2}}{4}}{- 1}$ .

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

Step 3.4.2.3.1.6

Rewrite $- 1 \cdot \frac{K^{2}}{4}$ as $- \frac{K^{2}}{4}$ .

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

Step 3.4.2.3.1.7

Divide $- 1$ by $- 1$ .

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

Step 4

Simplify the constant of integration.

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