Calculus Examples

$x (y d) x + \sqrt{1 + x^{2}} d y = 0$

Step 1

Subtract $x y d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $\frac{1}{\sqrt{1 + x^{2}} y}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $\sqrt{1 + x^{2}}$ .

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Step 3.1.1

Factor $\sqrt{1 + x^{2}}$ out of $\sqrt{1 + x^{2}} y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

Move the leading negative in $- \frac{1}{\sqrt{1 + x^{2}} y}$ into the numerator.

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

Step 3.3.2

Factor $y$ out of $\sqrt{1 + x^{2}} y$ .

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

Step 3.3.3

Factor $y$ out of $x y$ .

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

Step 3.3.4

Cancel the common factor.

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

Step 3.3.5

Rewrite the expression.

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

Step 3.4

Combine $\frac{- 1}{\sqrt{1 + x^{2}}}$ and $x$ .

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

Step 3.5

Move the negative in front of the fraction.

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

Step 3.6

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 3.7

Combine and simplify the denominator.

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Step 3.7.1

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 3.7.2

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 3.7.3

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 3.7.4

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

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

Step 3.7.5

Add $1$ and $1$ .

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

Step 3.7.6

Rewrite ${\sqrt{1 + x^{2}}}^{2}$ as $1 + x^{2}$ .

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Step 3.7.6.1

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

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

Step 3.7.6.2

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

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

Step 3.7.6.3

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

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

Step 3.7.6.4

Cancel the common factor of $2$ .

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Step 3.7.6.4.1

Cancel the common factor.

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

Step 3.7.6.4.2

Rewrite the expression.

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

Step 3.7.6.5

Simplify.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

Let $u = 1 + x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.2.1

Let $u = 1 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 4.3.2.1.1

Differentiate $1 + x^{2}$ .

$\frac{d}{d x} [1 + x^{2}]$

Step 4.3.2.1.2

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$

Step 4.3.2.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 4.3.2.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 + 2 x$

Step 4.3.2.1.5

Add $0$ and $2 x$ .

$2 x$

Step 4.3.2.2

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

$\ln (| y |) + C_{1} = - \int \frac{\sqrt{u}}{u} \cdot \frac{1}{2} d u$

Step 4.3.3

Simplify.

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Step 4.3.3.1

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

$\ln (| y |) + C_{1} = - \int \frac{\sqrt{u}}{u \cdot 2} d u$

Step 4.3.3.2

Move $2$ to the left of $u$ .

$\ln (| y |) + C_{1} = - \int \frac{\sqrt{u}}{2 u} d u$

Step 4.3.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\ln (| y |) + C_{1} = - (\frac{1}{2} \int \frac{\sqrt{u}}{u} d u)$

Step 4.3.5

Simplify the expression.

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Step 4.3.5.1

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

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

Step 4.3.5.2

Simplify.

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Step 4.3.5.2.1

Move $u^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\ln (| y |) + C_{1} = - \frac{1}{2} \int \frac{1}{u \cdot u^{- \frac{1}{2}}} d u$

Step 4.3.5.2.2

Multiply $u$ by $u^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.3.5.2.2.1

Multiply $u$ by $u^{- \frac{1}{2}}$ .

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Step 4.3.5.2.2.1.1

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

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

Step 4.3.5.2.2.1.2

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

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

Step 4.3.5.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 4.3.5.2.2.3

Combine the numerators over the common denominator.

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

Step 4.3.5.2.2.4

Subtract $1$ from $2$ .

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

Step 4.3.5.3

Apply basic rules of exponents.

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Step 4.3.5.3.1

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

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

Step 4.3.5.3.2

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

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Step 4.3.5.3.2.1

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

$\ln (| y |) + C_{1} = - \frac{1}{2} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.5.3.2.2

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

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

Step 4.3.5.3.2.3

Move the negative in front of the fraction.

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

Step 4.3.6

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

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

Step 4.3.7

Simplify.

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Step 4.3.7.1

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

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

Step 4.3.7.2

Simplify.

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Step 4.3.7.2.1

Multiply $2$ by $- 1$ .

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

Step 4.3.7.2.2

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

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

Step 4.3.7.2.3

Cancel the common factor of $- 2$ and $2$ .

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Step 4.3.7.2.3.1

Factor $2$ out of $- 2$ .

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

Step 4.3.7.2.3.2

Cancel the common factors.

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Step 4.3.7.2.3.2.1

Factor $2$ out of $2$ .

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

Step 4.3.7.2.3.2.2

Cancel the common factor.

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

Step 4.3.7.2.3.2.3

Rewrite the expression.

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

Step 4.3.7.2.3.2.4

Divide $- 1$ by $1$ .

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

Step 4.3.8

Replace all occurrences of $u$ with $1 + x^{2}$ .

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

To solve for $y$ , rewrite the equation using properties of logarithms.

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

Step 5.2

Rewrite $\ln (| y |) = - {(1 + x^{2})}^{\frac{1}{2}} + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

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

Step 5.3

Solve for $y$ .

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Step 5.3.1

Rewrite the equation as $| y | = e^{- {(1 + x^{2})}^{\frac{1}{2}} + C}$ .

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

Step 5.3.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

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

Step 6

Group the constant terms together.

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Step 6.1

Rewrite $e^{- {(1 + x^{2})}^{\frac{1}{2}} + C}$ as $e^{- {(1 + x^{2})}^{\frac{1}{2}}} e^{C}$ .

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

Step 6.2

Reorder $e^{- {(1 + x^{2})}^{\frac{1}{2}}}$ and $e^{C}$ .

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

Step 6.3

Combine constants with the plus or minus.

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