Calculus Examples

$\int \sqrt{3 - 2 x - x^{2}} d x$

Step 1

Complete the square.

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Step 1.1

Simplify the expression.

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Step 1.1.1

Move $3$ .

$- 2 x - x^{2} + 3$

Step 1.1.2

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

$- x^{2} - 2 x + 3$

Step 1.2

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

$a = - 1$

$b = - 2$

$c = 3$

Step 1.3

Consider the vertex form of a parabola.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

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

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Step 1.4.2.1.1

Factor $2$ out of $- 2$ .

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

Step 1.4.2.1.2

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

$d = - 1 \cdot - 1$

Step 1.4.2.2

Rewrite $- 1 \cdot - 1$ as $- - 1$ .

$d = - - 1$

Step 1.4.2.3

Multiply $- 1$ by $- 1$ .

$d = 1$

Step 1.5

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

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Step 1.5.1

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

$e = 3 - \frac{{(- 2)}^{2}}{4 \cdot - 1}$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

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

$e = 3 - \frac{4}{4 \cdot - 1}$

Step 1.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 3 - \frac{4}{- 4}$

Step 1.5.2.1.3

Divide $4$ by $- 4$ .

$e = 3 - - 1$

Step 1.5.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 3 + 1$

Step 1.5.2.2

Add $3$ and $1$ .

$e = 4$

Step 1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 1)}^{2} + 4$ .

$\int \sqrt{- {(x + 1)}^{2} + 4} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $x + 1$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

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

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$\int \sqrt{- {u_{1}}^{2} + 4} d u_{1}$

Step 3

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

$\int \sqrt{- {(2 \sin (t))}^{2} + 4} (2 \cos (t)) d t$

Step 4

Simplify terms.

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Step 4.1

Simplify $\sqrt{- {(2 \sin (t))}^{2} + 4}$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $2 \sin (t)$ .

$\int \sqrt{- (2^{2} \sin^{2} (t)) + 4} (2 \cos (t)) d t$

Step 4.1.1.2

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

$\int \sqrt{- (4 \sin^{2} (t)) + 4} (2 \cos (t)) d t$

Step 4.1.1.3

Multiply $4$ by $- 1$ .

$\int \sqrt{- 4 \sin^{2} (t) + 4} (2 \cos (t)) d t$

Step 4.1.2

Reorder $- 4 \sin^{2} (t)$ and $4$ .

$\int \sqrt{4 - 4 \sin^{2} (t)} (2 \cos (t)) d t$

Step 4.1.3

Factor $4$ out of $4$ .

$\int \sqrt{4 (1) - 4 \sin^{2} (t)} (2 \cos (t)) d t$

Step 4.1.4

Factor $4$ out of $- 4 \sin^{2} (t)$ .

$\int \sqrt{4 (1) + 4 (- \sin^{2} (t))} (2 \cos (t)) d t$

Step 4.1.5

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

$\int \sqrt{4 (1 - \sin^{2} (t))} (2 \cos (t)) d t$

Step 4.1.6

Apply pythagorean identity.

$\int \sqrt{4 \cos^{2} (t)} (2 \cos (t)) d t$

Step 4.1.7

Rewrite $4 \cos^{2} (t)$ as ${(2 \cos (t))}^{2}$ .

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

Step 4.1.8

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

$\int 2 \cos (t) (2 \cos (t)) d t$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $2$ by $2$ .

$\int 4 \cos (t) \cos (t) d t$

Step 4.2.2

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

$\int 4 (\cos^{1} (t) \cos (t)) d t$

Step 4.2.3

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

$\int 4 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 4.2.4

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

$\int 4 {\cos (t)}^{1 + 1} d t$

Step 4.2.5

Add $1$ and $1$ .

$\int 4 \cos^{2} (t) d t$

Step 5

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$4 \int \cos^{2} (t) d t$

Step 6

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

$4 \int \frac{1 + \cos (2 t)}{2} d t$

Step 7

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

$4 (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} \int 1 + \cos (2 t) d t$

Step 8.2

Cancel the common factor of $4$ and $2$ .

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Step 8.2.1

Factor $2$ out of $4$ .

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

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.2.2

Cancel the common factor.

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

Step 8.2.2.3

Rewrite the expression.

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

Step 8.2.2.4

Divide $2$ by $1$ .

$2 \int 1 + \cos (2 t) d t$

Step 9

Split the single integral into multiple integrals.

$2 (\int d t + \int \cos (2 t) d t)$

Step 10

Apply the constant rule.

$2 (t + C + \int \cos (2 t) d t)$

Step 11

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 11.1

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

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Step 11.1.1

Differentiate $2 t$ .

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

Step 11.1.2

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 11.1.3

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 11.1.4

Multiply $2$ by $1$ .

$2$

Step 11.2

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Simplify.

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

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

Replace all occurrences of $t$ with $\arcsin (\frac{u_{1}}{2})$ .

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

Replace all occurrences of $t$ with $\arcsin (\frac{u_{1}}{2})$ .

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

Step 16.5

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

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

Step 17

Simplify.

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Step 17.1

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

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

Step 17.2

Apply the distributive property.

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

Step 17.3

Cancel the common factor of $2$ .

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Step 17.3.1

Cancel the common factor.

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

Step 17.3.2

Rewrite the expression.

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

Step 18

Reorder terms.

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