Calculus Examples

$\sqrt{9 - x^{2}}$

Step 1

Write $\sqrt{9 - x^{2}}$ as a function.

$f (x) = \sqrt{9 - x^{2}}$

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.

$F (x) = \int \sqrt{9 - x^{2}} d x$

Step 4

Let $x = 3 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 3 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $3 \cos (t)$ is positive.

$\int \sqrt{9 - {(3 \sin (t))}^{2}} (3 \cos (t)) d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{9 - {(3 \sin (t))}^{2}}$ .

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $3 \sin (t)$ .

$\int \sqrt{9 - (3^{2} \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.1.2

Raise $3$ to the power of $2$ .

$\int \sqrt{9 - (9 \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.1.3

Multiply $9$ by $- 1$ .

$\int \sqrt{9 - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 5.1.2

Factor $9$ out of $9$ .

$\int \sqrt{9 (1) - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 5.1.3

Factor $9$ out of $- 9 \sin^{2} (t)$ .

$\int \sqrt{9 (1) + 9 (- \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.4

Factor $9$ out of $9 (1) + 9 (- \sin^{2} (t))$ .

$\int \sqrt{9 (1 - \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.5

Apply pythagorean identity.

$\int \sqrt{9 \cos^{2} (t)} (3 \cos (t)) d t$

Step 5.1.6

Rewrite $9 \cos^{2} (t)$ as ${(3 \cos (t))}^{2}$ .

$\int \sqrt{{(3 \cos (t))}^{2}} (3 \cos (t)) d t$

Step 5.1.7

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

$\int 3 \cos (t) (3 \cos (t)) d t$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $3$ by $3$ .

$\int 9 \cos (t) \cos (t) d t$

Step 5.2.2

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

$\int 9 (\cos^{1} (t) \cos (t)) d t$

Step 5.2.3

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

$\int 9 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 5.2.4

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

$\int 9 {\cos (t)}^{1 + 1} d t$

Step 5.2.5

Add $1$ and $1$ .

$\int 9 \cos^{2} (t) d t$

Step 6

Since $9$ is constant with respect to $t$ , move $9$ out of the integral.

$9 \int \cos^{2} (t) d t$

Step 7

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

$9 \int \frac{1 + \cos (2 t)}{2} d t$

Step 8

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

$9 (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 9

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} \int 1 + \cos (2 t) d t$

Step 10

Split the single integral into multiple integrals.

$\frac{9}{2} (\int d t + \int \cos (2 t) d t)$

Step 11

Apply the constant rule.

$\frac{9}{2} (t + C + \int \cos (2 t) d t)$

Step 12

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

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Step 12.1

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

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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 $u$ and $d u$ .

$\frac{9}{2} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 13

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

$\frac{9}{2} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 14

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

$\frac{9}{2} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 15

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{9}{2} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 16

Simplify.

$\frac{9}{2} (t + \frac{1}{2} \sin (u)) + C$

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

Replace all occurrences of $t$ with $\arcsin (\frac{x}{3})$ .

$\frac{9}{2} (\arcsin (\frac{x}{3}) + \frac{1}{2} \sin (u)) + C$

Step 17.2

Replace all occurrences of $u$ with $2 t$ .

$\frac{9}{2} (\arcsin (\frac{x}{3}) + \frac{1}{2} \sin (2 t)) + C$

Step 17.3

Replace all occurrences of $t$ with $\arcsin (\frac{x}{3})$ .

$\frac{9}{2} (\arcsin (\frac{x}{3}) + \frac{1}{2} \sin (2 \arcsin (\frac{x}{3}))) + C$

Step 18

Simplify.

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Step 18.1

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (\frac{x}{3}))$ .

$\frac{9}{2} (\arcsin (\frac{x}{3}) + \frac{\sin (2 \arcsin (\frac{x}{3}))}{2}) + C$

Step 18.2

Apply the distributive property.

$\frac{9}{2} \arcsin (\frac{x}{3}) + \frac{9}{2} \cdot \frac{\sin (2 \arcsin (\frac{x}{3}))}{2} + C$

Step 18.3

Combine $\frac{9}{2}$ and $\arcsin (\frac{x}{3})$ .

$\frac{9 \arcsin (\frac{x}{3})}{2} + \frac{9}{2} \cdot \frac{\sin (2 \arcsin (\frac{x}{3}))}{2} + C$

Step 18.4

Multiply $\frac{9}{2} \cdot \frac{\sin (2 \arcsin (\frac{x}{3}))}{2}$ .

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Step 18.4.1

Multiply $\frac{9}{2}$ by $\frac{\sin (2 \arcsin (\frac{x}{3}))}{2}$ .

$\frac{9 \arcsin (\frac{x}{3})}{2} + \frac{9 \sin (2 \arcsin (\frac{x}{3}))}{2 \cdot 2} + C$

Step 18.4.2

Multiply $2$ by $2$ .

$\frac{9 \arcsin (\frac{x}{3})}{2} + \frac{9 \sin (2 \arcsin (\frac{x}{3}))}{4} + C$

Step 19

Reorder terms.

$\frac{9}{2} \arcsin (\frac{1}{3} x) + \frac{9}{4} \sin (2 \arcsin (\frac{1}{3} x)) + C$

Step 20

The answer is the antiderivative of the function $f (x) = \sqrt{9 - x^{2}}$ .

$F (x) =$ $\frac{9}{2} \arcsin (\frac{1}{3} x) + \frac{9}{4} \sin (2 \arcsin (\frac{1}{3} x)) + C$