Calculus Examples

$\int_{- k}^{k} \sqrt{k^{2} - x^{2}} d x$

Step 1

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

$\frac{d}{d k} [\int_{- k}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 2

Split the integral into two integrals where $c$ is some value between $- k$ and $k$ .

$\frac{d}{d k} [\int_{- k}^{c} {(k^{2} - x^{2})}^{\frac{1}{2}} d x + \int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 3

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

$\frac{d}{d k} [\int_{- k}^{c} {(k^{2} - x^{2})}^{\frac{1}{2}} d x] + \frac{d}{d k} [\int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 4

Swap the bounds of integration.

$\frac{d}{d k} [- \int_{c}^{- k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x] + \frac{d}{d k} [\int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 5

Take the derivative of $- \int_{c}^{- k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x$ with respect to $k$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d k} [- k] (- {(k^{2} - {(- k)}^{2})}^{\frac{1}{2}}) + \frac{d}{d k} [\int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 6

Differentiate.

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Step 6.1

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

$- \frac{d}{d k} [k] (- {(k^{2} - {(- k)}^{2})}^{\frac{1}{2}}) + \frac{d}{d k} [\int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 6.2

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

$- 1 \cdot 1 (- {(k^{2} - {(- k)}^{2})}^{\frac{1}{2}}) + \frac{d}{d k} [\int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 6.3

Multiply $- 1$ by $1$ .

$- - {(k^{2} - {(- k)}^{2})}^{\frac{1}{2}} + \frac{d}{d k} [\int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x]$

Step 7

Take the derivative of $\int_{c}^{k} {(k^{2} - x^{2})}^{\frac{1}{2}} d x$ with respect to $k$ using Fundamental Theorem of Calculus.

$- - {(k^{2} - {(- k)}^{2})}^{\frac{1}{2}} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8

Simplify terms.

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Step 8.1

Factor $- 1$ out of $- k$ .

$- - {(k^{2} - {(- (k))}^{2})}^{\frac{1}{2}} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Apply the product rule to $- (k)$ .

$- - {(k^{2} - ({(- 1)}^{2} k^{2}))}^{\frac{1}{2}} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.2.2

Raise $- 1$ to the power of $2$ .

$- - {(k^{2} - (1 k^{2}))}^{\frac{1}{2}} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.2.3

Multiply $k^{2}$ by $1$ .

$- - {(k^{2} - k^{2})}^{\frac{1}{2}} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.3

Subtract $k^{2}$ from $k^{2}$ .

$- - 0^{\frac{1}{2}} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.4

Simplify the expression.

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Step 8.4.1

Rewrite $0$ as $0^{2}$ .

$- - {(0^{2})}^{\frac{1}{2}} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.4.2

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

$- - 0^{2 (\frac{1}{2})} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.5

Cancel the common factor of $2$ .

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Step 8.5.1

Cancel the common factor.

$- - 0^{2 (\frac{1}{2})} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.5.2

Rewrite the expression.

$- - 0^{1} + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.6

Evaluate the exponent.

$- - 0 + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.7

Multiply by zero.

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Step 8.7.1

Multiply $- 1$ by $0$ .

$- 0 + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.7.2

Multiply $- 1$ by $0$ .

$0 + {(k^{2} - k^{2})}^{\frac{1}{2}}$

Step 8.8

Subtract $k^{2}$ from $k^{2}$ .

$0 + 0^{\frac{1}{2}}$

Step 8.9

Simplify the expression.

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Step 8.9.1

Rewrite $0$ as $0^{2}$ .

$0 + {(0^{2})}^{\frac{1}{2}}$

Step 8.9.2

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

$0 + 0^{2 (\frac{1}{2})}$

Step 8.10

Cancel the common factor of $2$ .

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Step 8.10.1

Cancel the common factor.

$0 + 0^{2 (\frac{1}{2})}$

Step 8.10.2

Rewrite the expression.

$0 + 0^{1}$

Step 8.11

Evaluate the exponent.

$0 + 0$

Step 8.12

Add $0$ and $0$ .

$0$