Calculus Examples

$x d x + \sqrt{a^{2} - x^{2}} d y = 0$

Step 1

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

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

Step 2

Multiply both sides by $\frac{1}{\sqrt{a^{2} - x^{2}}}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $\sqrt{a^{2} - x^{2}}$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = x$ .

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

Step 3.4

Multiply $\frac{1}{\sqrt{(a + x) (a - x)}}$ by $\frac{\sqrt{(a + x) (a - x)}}{\sqrt{(a + x) (a - x)}}$ .

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

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{1}{\sqrt{(a + x) (a - x)}}$ by $\frac{\sqrt{(a + x) (a - x)}}{\sqrt{(a + x) (a - x)}}$ .

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

Step 3.5.2

Raise $\sqrt{(a + x) (a - x)}$ to the power of $1$ .

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

Step 3.5.3

Raise $\sqrt{(a + x) (a - x)}$ to the power of $1$ .

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

Step 3.5.4

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

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

Step 3.5.5

Add $1$ and $1$ .

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

Step 3.5.6

Rewrite ${\sqrt{(a + x) (a - x)}}^{2}$ as $(a + x) (a - x)$ .

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Step 3.5.6.1

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

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

Step 3.5.6.2

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

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

Step 3.5.6.3

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

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

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

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

Step 3.5.6.4.2

Rewrite the expression.

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

Step 3.5.6.5

Simplify.

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

Step 3.6

Combine $x$ and $\frac{\sqrt{(a + x) (a - x)}}{(a + x) (a - x)}$ .

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int d y = \int - \frac{x \sqrt{(a + x) (a - x)}}{(a + x) (a - x)} d x$

Step 4.2

Apply the constant rule.

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

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.

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

Step 4.3.2

Let $u = (a + x) (a - x)$ . 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 $u = (a + x) (a - x)$ . Find $\frac{d u}{d x}$ .

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Step 4.3.2.1.1

Differentiate $(a + x) (a - x)$ .

$\frac{d}{d x} [(a + x) (a - x)]$

Step 4.3.2.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = a + x$ and $g (x) = a - x$ .

$(a + x) \frac{d}{d x} [a - x] + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3

Differentiate.

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Step 4.3.2.1.3.1

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

$(a + x) (\frac{d}{d x} [a] + \frac{d}{d x} [- x]) + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.2

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

$(a + x) (0 + \frac{d}{d x} [- x]) + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$(a + x) \frac{d}{d x} [- x] + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$(a + x) (- \frac{d}{d x} [x]) + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.5

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

$(a + x) (- 1 \cdot 1) + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.6

Simplify the expression.

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Step 4.3.2.1.3.6.1

Multiply $- 1$ by $1$ .

$(a + x) \cdot - 1 + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.6.2

Move $- 1$ to the left of $a + x$ .

$- 1 \cdot (a + x) + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.6.3

Rewrite $- 1 (a + x)$ as $- (a + x)$ .

$- (a + x) + (a - x) \frac{d}{d x} [a + x]$

Step 4.3.2.1.3.7

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

$- (a + x) + (a - x) (\frac{d}{d x} [a] + \frac{d}{d x} [x])$

Step 4.3.2.1.3.8

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

$- (a + x) + (a - x) (0 + \frac{d}{d x} [x])$

Step 4.3.2.1.3.9

Add $0$ and $\frac{d}{d x} [x]$ .

$- (a + x) + (a - x) \frac{d}{d x} [x]$

Step 4.3.2.1.3.10

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

$- (a + x) + (a - x) \cdot 1$

Step 4.3.2.1.3.11

Multiply $a - x$ by $1$ .

$- (a + x) + a - x$

Step 4.3.2.1.4

Simplify.

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Step 4.3.2.1.4.1

Apply the distributive property.

$- a - x + a - x$

Step 4.3.2.1.4.2

Combine terms.

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Step 4.3.2.1.4.2.1

Add $- a$ and $a$ .

$- x + 0 - x$

Step 4.3.2.1.4.2.2

Add $- x$ and $0$ .

$- x - x$

Step 4.3.2.1.4.2.3

Subtract $x$ from $- x$ .

$- 2 x$

Step 4.3.2.2

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

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

Step 4.3.3

Simplify.

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Step 4.3.3.1

Move the negative in front of the fraction.

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

Step 4.3.3.2

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

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

Step 4.3.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

Simplify.

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Step 4.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.5.2

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

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

Step 4.3.6

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

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

Step 4.3.7

Simplify the expression.

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Step 4.3.7.1

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

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

Step 4.3.7.2

Simplify.

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Step 4.3.7.2.1

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

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

Step 4.3.7.2.2

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

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Step 4.3.7.2.2.1

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

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Step 4.3.7.2.2.1.1

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

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

Step 4.3.7.2.2.1.2

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

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

Step 4.3.7.2.2.2

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

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

Step 4.3.7.2.2.3

Combine the numerators over the common denominator.

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

Step 4.3.7.2.2.4

Subtract $1$ from $2$ .

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

Step 4.3.7.3

Apply basic rules of exponents.

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Step 4.3.7.3.1

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

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

Step 4.3.7.3.2

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

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Step 4.3.7.3.2.1

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

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

Step 4.3.7.3.2.2

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

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

Step 4.3.7.3.2.3

Move the negative in front of the fraction.

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

Step 4.3.8

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

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

Step 4.3.9

Simplify.

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Step 4.3.9.1

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

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

Step 4.3.9.2

Simplify.

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Step 4.3.9.2.1

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

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

Step 4.3.9.2.2

Cancel the common factor of $2$ .

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Step 4.3.9.2.2.1

Cancel the common factor.

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

Step 4.3.9.2.2.2

Rewrite the expression.

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

Step 4.3.9.2.3

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

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

Step 4.3.10

Replace all occurrences of $u$ with $(a + x) (a - x)$ .

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

Step 4.4

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

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