Calculus Examples

$\int \sqrt{16 - 9 x^{2}} d x$

Step 1

Let $x = \frac{4}{3} \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{4 \cos (t)}{3} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{4 \cos (t)}{3}$ is positive.

$\int \sqrt{16 - 9 {(\frac{4}{3} \sin (t))}^{2}} \frac{4 \cos (t)}{3} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{16 - 9 {(\frac{4}{3} \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{4}{3}$ and $\sin (t)$ .

$\int \sqrt{16 - 9 {(\frac{4 \sin (t)}{3})}^{2}} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.2.1

Apply the product rule to $\frac{4 \sin (t)}{3}$ .

$\int \sqrt{16 - 9 \frac{{(4 \sin (t))}^{2}}{3^{2}}} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.2.2

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

$\int \sqrt{16 - 9 \frac{4^{2} \sin^{2} (t)}{3^{2}}} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.3

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

$\int \sqrt{16 - 9 \frac{16 \sin^{2} (t)}{3^{2}}} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.4

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

$\int \sqrt{16 - 9 \frac{16 \sin^{2} (t)}{9}} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.5

Cancel the common factor of $9$ .

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Step 2.1.1.5.1

Factor $9$ out of $- 9$ .

$\int \sqrt{16 + 9 (- 1) \frac{16 \sin^{2} (t)}{9}} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.5.2

Cancel the common factor.

$\int \sqrt{16 + 9 \cdot - 1 \frac{16 \sin^{2} (t)}{9}} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.5.3

Rewrite the expression.

$\int \sqrt{16 - 1 (16 \sin^{2} (t))} \frac{4 \cos (t)}{3} d t$

Step 2.1.1.6

Multiply $16$ by $- 1$ .

$\int \sqrt{16 - 16 \sin^{2} (t)} \frac{4 \cos (t)}{3} d t$

Step 2.1.2

Factor $16$ out of $16$ .

$\int \sqrt{16 (1) - 16 \sin^{2} (t)} \frac{4 \cos (t)}{3} d t$

Step 2.1.3

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

$\int \sqrt{16 (1) + 16 (- \sin^{2} (t))} \frac{4 \cos (t)}{3} d t$

Step 2.1.4

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

$\int \sqrt{16 (1 - \sin^{2} (t))} \frac{4 \cos (t)}{3} d t$

Step 2.1.5

Apply pythagorean identity.

$\int \sqrt{16 \cos^{2} (t)} \frac{4 \cos (t)}{3} d t$

Step 2.1.6

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

$\int \sqrt{{(4 \cos (t))}^{2}} \frac{4 \cos (t)}{3} d t$

Step 2.1.7

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

$\int 4 \cos (t) \frac{4 \cos (t)}{3} d t$

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{4 \cos (t)}{3}$ and $4$ .

$\int \frac{4 \cos (t) \cdot 4}{3} \cos (t) d t$

Step 2.2.2

Multiply $4$ by $4$ .

$\int \frac{16 \cos (t)}{3} \cos (t) d t$

Step 2.2.3

Combine $\frac{16 \cos (t)}{3}$ and $\cos (t)$ .

$\int \frac{16 \cos (t) \cos (t)}{3} d t$

Step 2.2.4

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

$\int \frac{16 (\cos^{1} (t) \cos (t))}{3} d t$

Step 2.2.5

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

$\int \frac{16 (\cos^{1} (t) \cos^{1} (t))}{3} d t$

Step 2.2.6

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

$\int \frac{16 {\cos (t)}^{1 + 1}}{3} d t$

Step 2.2.7

Add $1$ and $1$ .

$\int \frac{16 \cos^{2} (t)}{3} d t$

Step 3

Since $\frac{16}{3}$ is constant with respect to $t$ , move $\frac{16}{3}$ out of the integral.

$\frac{16}{3} \int \cos^{2} (t) d t$

Step 4

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

$\frac{16}{3} \int \frac{1 + \cos (2 t)}{2} d t$

Step 5

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

$\frac{16}{3} (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1}{2}$ by $\frac{16}{3}$ .

$\frac{16}{2 \cdot 3} \int 1 + \cos (2 t) d t$

Step 6.2

Multiply $2$ by $3$ .

$\frac{16}{6} \int 1 + \cos (2 t) d t$

Step 6.3

Cancel the common factor of $16$ and $6$ .

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Step 6.3.1

Factor $2$ out of $16$ .

$\frac{2 (8)}{6} \int 1 + \cos (2 t) d t$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 8}{2 \cdot 3} \int 1 + \cos (2 t) d t$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 \cdot 3} \int 1 + \cos (2 t) d t$

Step 6.3.2.3

Rewrite the expression.

$\frac{8}{3} \int 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$\frac{8}{3} (\int d t + \int \cos (2 t) d t)$

Step 8

Apply the constant rule.

$\frac{8}{3} (t + C + \int \cos (2 t) d t)$

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 $d u$ .

$\frac{8}{3} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 10

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

$\frac{8}{3} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 11

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

$\frac{8}{3} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 12

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

$\frac{8}{3} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 13

Simplify.

$\frac{8}{3} (t + \frac{1}{2} \sin (u)) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $t$ with $\arcsin (\frac{3 x}{4})$ .

$\frac{8}{3} (\arcsin (\frac{3 x}{4}) + \frac{1}{2} \sin (u)) + C$

Step 14.2

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

$\frac{8}{3} (\arcsin (\frac{3 x}{4}) + \frac{1}{2} \sin (2 t)) + C$

Step 14.3

Replace all occurrences of $t$ with $\arcsin (\frac{3 x}{4})$ .

$\frac{8}{3} (\arcsin (\frac{3 x}{4}) + \frac{1}{2} \sin (2 \arcsin (\frac{3 x}{4}))) + C$

Step 15

Simplify.

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Step 15.1

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (\frac{3 x}{4}))$ .

$\frac{8}{3} (\arcsin (\frac{3 x}{4}) + \frac{\sin (2 \arcsin (\frac{3 x}{4}))}{2}) + C$

Step 15.2

Apply the distributive property.

$\frac{8}{3} \arcsin (\frac{3 x}{4}) + \frac{8}{3} \cdot \frac{\sin (2 \arcsin (\frac{3 x}{4}))}{2} + C$

Step 15.3

Combine $\frac{8}{3}$ and $\arcsin (\frac{3 x}{4})$ .

$\frac{8 \arcsin (\frac{3 x}{4})}{3} + \frac{8}{3} \cdot \frac{\sin (2 \arcsin (\frac{3 x}{4}))}{2} + C$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Factor $2$ out of $8$ .

$\frac{8 \arcsin (\frac{3 x}{4})}{3} + \frac{2 (4)}{3} \cdot \frac{\sin (2 \arcsin (\frac{3 x}{4}))}{2} + C$

Step 15.4.2

Cancel the common factor.

$\frac{8 \arcsin (\frac{3 x}{4})}{3} + \frac{2 \cdot 4}{3} \cdot \frac{\sin (2 \arcsin (\frac{3 x}{4}))}{2} + C$

Step 15.4.3

Rewrite the expression.

$\frac{8 \arcsin (\frac{3 x}{4})}{3} + \frac{4}{3} \sin (2 \arcsin (\frac{3 x}{4})) + C$

Step 15.5

Combine $\frac{4}{3}$ and $\sin (2 \arcsin (\frac{3 x}{4}))$ .

$\frac{8 \arcsin (\frac{3 x}{4})}{3} + \frac{4 \sin (2 \arcsin (\frac{3 x}{4}))}{3} + C$

Step 16

Reorder terms.

$\frac{8}{3} \arcsin (\frac{3}{4} x) + \frac{4}{3} \sin (2 \arcsin (\frac{3}{4} x)) + C$