Calculus Examples

$\int \sqrt{16 - x^{2}} d x$

Step 1

Let $x = 4 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 4 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $4 \cos (t)$ is positive.

$\int \sqrt{16 - {(4 \sin (t))}^{2}} (4 \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{16 - {(4 \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $4 \sin (t)$ .

$\int \sqrt{16 - (4^{2} \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.1.2

Raise $4$ to the power of $2$ .

$\int \sqrt{16 - (16 \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.1.3

Multiply $16$ by $- 1$ .

$\int \sqrt{16 - 16 \sin^{2} (t)} (4 \cos (t)) d t$

Step 2.1.2

Factor $16$ out of $16$ .

$\int \sqrt{16 (1) - 16 \sin^{2} (t)} (4 \cos (t)) d t$

Step 2.1.3

Factor $16$ out of $- 16 \sin^{2} (t)$ .

$\int \sqrt{16 (1) + 16 (- \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.4

Factor $16$ out of $16 (1) + 16 (- \sin^{2} (t))$ .

$\int \sqrt{16 (1 - \sin^{2} (t))} (4 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int \sqrt{16 \cos^{2} (t)} (4 \cos (t)) d t$

Step 2.1.6

Rewrite $16 \cos^{2} (t)$ as ${(4 \cos (t))}^{2}$ .

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

Step 2.1.7

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

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

Step 2.2

Simplify.

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Step 2.2.1

Multiply $4$ by $4$ .

$\int 16 \cos (t) \cos (t) d t$

Step 2.2.2

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

$\int 16 (\cos^{1} (t) \cos (t)) d t$

Step 2.2.3

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

$\int 16 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 2.2.4

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

$\int 16 {\cos (t)}^{1 + 1} d t$

Step 2.2.5

Add $1$ and $1$ .

$\int 16 \cos^{2} (t) d t$

Step 3

Since $16$ is constant with respect to $t$ , move $16$ out of the integral.

$16 \int \cos^{2} (t) d t$

Step 4

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

$16 \int \frac{1 + \cos (2 t)}{2} d t$

Step 5

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

$16 (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $16$ .

$\frac{16}{2} \int 1 + \cos (2 t) d t$

Step 6.2

Cancel the common factor of $16$ and $2$ .

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Step 6.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 8}{2} \int 1 + \cos (2 t) d t$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 8}{2 (1)} \int 1 + \cos (2 t) d t$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 \cdot 1} \int 1 + \cos (2 t) d t$

Step 6.2.2.3

Rewrite the expression.

$\frac{8}{1} \int 1 + \cos (2 t) d t$

Step 6.2.2.4

Divide $8$ by $1$ .

$8 \int 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$8 (\int d t + \int \cos (2 t) d t)$

Step 8

Apply the constant rule.

$8 (t + C + \int \cos (2 t) d t)$

Step 9

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 9.1

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

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Step 9.1.1

Differentiate $2 t$ .

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

Step 9.1.2

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 9.1.3

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 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

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

$8 (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 10

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

$8 (t + C + \int \frac{\cos (u)}{2} d u)$

Step 11

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

$8 (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 12

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

$8 (t + C + \frac{1}{2} (\sin (u) + C))$

Step 13

Simplify.

$8 (t + \frac{1}{2} \sin (u)) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $t$ with $\arcsin (\frac{x}{4})$ .

$8 (\arcsin (\frac{x}{4}) + \frac{1}{2} \sin (u)) + C$

Step 14.2

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

$8 (\arcsin (\frac{x}{4}) + \frac{1}{2} \sin (2 t)) + C$

Step 14.3

Replace all occurrences of $t$ with $\arcsin (\frac{x}{4})$ .

$8 (\arcsin (\frac{x}{4}) + \frac{1}{2} \sin (2 \arcsin (\frac{x}{4}))) + C$

Step 15

Simplify.

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Step 15.1

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (\frac{x}{4}))$ .

$8 (\arcsin (\frac{x}{4}) + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 15.2

Apply the distributive property.

$8 \arcsin (\frac{x}{4}) + 8 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 15.3

Cancel the common factor of $2$ .

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Step 15.3.1

Factor $2$ out of $8$ .

$8 \arcsin (\frac{x}{4}) + 2 (4) \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 15.3.2

Cancel the common factor.

$8 \arcsin (\frac{x}{4}) + 2 \cdot 4 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 15.3.3

Rewrite the expression.

$8 \arcsin (\frac{x}{4}) + 4 \sin (2 \arcsin (\frac{x}{4})) + C$

Step 16

Reorder terms.

$8 \arcsin (\frac{1}{4} x) + 4 \sin (2 \arcsin (\frac{1}{4} x)) + C$