Calculus Examples

$\sqrt{2 x - x^{2}}$

Step 1

Write $\sqrt{2 x - x^{2}}$ as a function.

$f (x) = \sqrt{2 x - x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sqrt{2 x - x^{2}} d x$

Step 4

Complete the square.

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Step 4.1

Reorder $2 x$ and $- x^{2}$ .

$- x^{2} + 2 x$

Step 4.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 2$

$c = 0$

Step 4.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 4.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 4.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot - 1}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.1.1

Cancel the common factor.

$d = \frac{2}{2 \cdot - 1}$

Step 4.4.2.1.2

Rewrite the expression.

$d = \frac{1}{- 1}$

Step 4.4.2.1.3

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

$d = - 1 \cdot 1$

Step 4.4.2.2

Multiply $- 1$ by $1$ .

$d = - 1$

Step 4.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 4.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{2^{2}}{4 \cdot - 1}$

Step 4.5.2

Simplify the right side.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

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

$e = 0 - \frac{4}{4 \cdot - 1}$

Step 4.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 0 - \frac{4}{- 4}$

Step 4.5.2.1.3

Divide $4$ by $- 4$ .

$e = 0 - - 1$

Step 4.5.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 0 + 1$

Step 4.5.2.2

Add $0$ and $1$ .

$e = 1$

Step 4.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 1)}^{2} + 1$ .

$\int \sqrt{- {(x - 1)}^{2} + 1} d x$

Step 5

Let $u_{1} = x - 1$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 5.1

Let $u_{1} = x - 1$ . Find $\frac{d u_{1}}{d x}$ .

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Step 5.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 5.1.2

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

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 5.1.3

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

$1 + \frac{d}{d x} [- 1]$

Step 5.1.4

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

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \sqrt{- {u_{1}}^{2} + 1} d u_{1}$

Step 6

Let $u_{1} = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$\int \sqrt{- \sin^{2} (t) + 1} \cos (t) d t$

Step 7

Simplify terms.

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Step 7.1

Simplify $\sqrt{- \sin^{2} (t) + 1}$ .

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Step 7.1.1

Reorder $- \sin^{2} (t)$ and $1$ .

$\int \sqrt{1 - \sin^{2} (t)} \cos (t) d t$

Step 7.1.2

Apply pythagorean identity.

$\int \sqrt{\cos^{2} (t)} \cos (t) d t$

Step 7.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\int \cos (t) \cos (t) d t$

Step 7.2

Simplify.

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Step 7.2.1

Raise $\cos (t)$ to the power of $1$ .

$\int \cos^{1} (t) \cos (t) d t$

Step 7.2.2

Raise $\cos (t)$ to the power of $1$ .

$\int \cos^{1} (t) \cos^{1} (t) d t$

Step 7.2.3

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

$\int {\cos (t)}^{1 + 1} d t$

Step 7.2.4

Add $1$ and $1$ .

$\int \cos^{2} (t) d t$

Step 8

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$\int \frac{1 + \cos (2 t)}{2} d t$

Step 9

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

$\frac{1}{2} \int 1 + \cos (2 t) d t$

Step 10

Split the single integral into multiple integrals.

$\frac{1}{2} (\int d t + \int \cos (2 t) d t)$

Step 11

Apply the constant rule.

$\frac{1}{2} (t + C + \int \cos (2 t) d t)$

Step 12

Let $u_{2} = 2 t$ . Then $d u_{2} = 2 d t$ , so $\frac{1}{2} d u_{2} = d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 12.1

Let $u_{2} = 2 t$ . Find $\frac{d u_{2}}{d t}$ .

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Step 12.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 12.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 12.1.3

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

$2 \cdot 1$

Step 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} (t + C + \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 13

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{2} (t + C + \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 14

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

$\frac{1}{2} (t + C + \frac{1}{2} \int \cos (u_{2}) d u_{2})$

Step 15

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{2} (t + C + \frac{1}{2} (\sin (u_{2}) + C))$

Step 16

Simplify.

$\frac{1}{2} (t + \frac{1}{2} \sin (u_{2})) + C$

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

Replace all occurrences of $t$ with $\arcsin (u_{1})$ .

$\frac{1}{2} (\arcsin (u_{1}) + \frac{1}{2} \sin (u_{2})) + C$

Step 17.2

Replace all occurrences of $u_{2}$ with $2 t$ .

$\frac{1}{2} (\arcsin (u_{1}) + \frac{1}{2} \sin (2 t)) + C$

Step 17.3

Replace all occurrences of $u_{1}$ with $x - 1$ .

$\frac{1}{2} (\arcsin (x - 1) + \frac{1}{2} \sin (2 t)) + C$

Step 17.4

Replace all occurrences of $t$ with $\arcsin (u_{1})$ .

$\frac{1}{2} (\arcsin (x - 1) + \frac{1}{2} \sin (2 \arcsin (u_{1}))) + C$

Step 17.5

Replace all occurrences of $u_{1}$ with $x - 1$ .

$\frac{1}{2} (\arcsin (x - 1) + \frac{1}{2} \sin (2 \arcsin (x - 1))) + C$

Step 18

Simplify.

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Step 18.1

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (x - 1))$ .

$\frac{1}{2} (\arcsin (x - 1) + \frac{\sin (2 \arcsin (x - 1))}{2}) + C$

Step 18.2

Apply the distributive property.

$\frac{1}{2} \arcsin (x - 1) + \frac{1}{2} \cdot \frac{\sin (2 \arcsin (x - 1))}{2} + C$

Step 18.3

Combine $\frac{1}{2}$ and $\arcsin (x - 1)$ .

$\frac{\arcsin (x - 1)}{2} + \frac{1}{2} \cdot \frac{\sin (2 \arcsin (x - 1))}{2} + C$

Step 18.4

Multiply $\frac{1}{2} \cdot \frac{\sin (2 \arcsin (x - 1))}{2}$ .

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Step 18.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 \arcsin (x - 1))}{2}$ .

$\frac{\arcsin (x - 1)}{2} + \frac{\sin (2 \arcsin (x - 1))}{2 \cdot 2} + C$

Step 18.4.2

Multiply $2$ by $2$ .

$\frac{\arcsin (x - 1)}{2} + \frac{\sin (2 \arcsin (x - 1))}{4} + C$

Step 19

Reorder terms.

$\frac{1}{2} \arcsin (x - 1) + \frac{1}{4} \sin (2 \arcsin (x - 1)) + C$

Step 20

The answer is the antiderivative of the function $f (x) = \sqrt{2 x - x^{2}}$ .

$F (x) =$ $\frac{1}{2} \arcsin (x - 1) + \frac{1}{4} \sin (2 \arcsin (x - 1)) + C$