Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x \frac{d y}{d x} = \sqrt{1 - y}$ by $x$ and simplify.

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Step 1.1.1

Divide each term in $x \frac{d y}{d x} = \sqrt{1 - y}$ by $x$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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 $d u$ .

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

Step 2.2.2

Move the negative in front of the fraction.

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

Step 2.2.3

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

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

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 $u^{\frac{1}{2}}$ .

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

Step 2.2.4.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.4.3.2

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

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

Step 2.2.4.3.3

Move the negative in front of the fraction.

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

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

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

Step 2.2.6.2

Multiply $- 1$ by $2$ .

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

Step 2.2.7

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

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

Step 2.3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 2.4

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

$- 2 {(1 - y)}^{\frac{1}{2}} = \ln (| x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

$\frac{- 2 {(1 - y)}^{\frac{1}{2}}}{- 2} = \frac{\ln (| x |)}{- 2} + \frac{C}{- 2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

$\frac{- 2 {(1 - y)}^{\frac{1}{2}}}{- 2} = \frac{\ln (| x |)}{- 2} + \frac{C}{- 2}$

Step 3.1.2.2

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

${(1 - y)}^{\frac{1}{2}} = \frac{\ln (| x |)}{- 2} + \frac{C}{- 2}$

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

Rewrite $\frac{\ln (| x |)}{- 2}$ as $\frac{1}{- 2} \ln (| x |)$ .

${(1 - y)}^{\frac{1}{2}} = \frac{1}{- 2} \ln (| x |) + \frac{C}{- 2}$

Step 3.1.3.1.2

Simplify $\frac{1}{- 2} \ln (| x |)$ by moving $- \frac{1}{- 2}$ inside the logarithm.

${(1 - y)}^{\frac{1}{2}} = - \ln ({| x |}^{- \frac{1}{- 2}}) + \frac{C}{- 2}$

Step 3.1.3.1.3

Move the negative in front of the fraction.

${(1 - y)}^{\frac{1}{2}} = - \ln ({| x |}^{- - \frac{1}{2}}) + \frac{C}{- 2}$

Step 3.1.3.1.4

Multiply $- - \frac{1}{2}$ .

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Step 3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

${(1 - y)}^{\frac{1}{2}} = - \ln ({| x |}^{1 (\frac{1}{2})}) + \frac{C}{- 2}$

Step 3.1.3.1.4.2

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

${(1 - y)}^{\frac{1}{2}} = - \ln ({| x |}^{\frac{1}{2}}) + \frac{C}{- 2}$

Step 3.1.3.1.5

Move the negative in front of the fraction.

${(1 - y)}^{\frac{1}{2}} = - \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2}$

Step 3.2

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

${({(1 - y)}^{\frac{1}{2}})}^{2} = {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}$

Step 3.3

Simplify the left side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

${(1 - y)}^{\frac{1}{2} \cdot 2} = {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}$

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

${(1 - y)}^{\frac{1}{2} \cdot 2} = {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}$

Step 3.3.1.1.2.2

Rewrite the expression.

${(1 - y)}^{1} = {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}$

Step 3.3.1.2

Simplify.

$1 - y = {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $1$ from both sides of the equation.

$- y = {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2} - 1$

Step 3.4.2

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

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Step 3.4.2.1

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

$\frac{- y}{- 1} = \frac{{(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}}{- 1} + \frac{- 1}{- 1}$

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.

$\frac{y}{1} = \frac{{(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}}{- 1} + \frac{- 1}{- 1}$

Step 3.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{{(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}}{- 1} + \frac{- 1}{- 1}$

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 $\frac{{(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}}{- 1}$ .

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

Step 3.4.2.3.1.2

Rewrite $- 1 \cdot {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}$ as $- {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2}$ .

$y = - {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2} + \frac{- 1}{- 1}$

Step 3.4.2.3.1.3

Divide $- 1$ by $- 1$ .

$y = - {(- \ln ({| x |}^{\frac{1}{2}}) - \frac{C}{2})}^{2} + 1$

Step 4

Simplify the constant of integration.

$y = - {(- \ln ({| x |}^{\frac{1}{2}}) + C)}^{2} + 1$