Calculus Examples

$\int \sqrt{8 - 3 x^{2}} d x$

Step 1

Let $x = \frac{2 \sqrt{2}}{\sqrt{3}} \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{2 \sqrt{6} \cos (t)}{3} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{2 \sqrt{6} \cos (t)}{3}$ is positive.

$\int \sqrt{{(2 \sqrt{2})}^{2} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2

Simplify terms.

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Step 2.1

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

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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 $2 \sqrt{2}$ .

$\int \sqrt{2^{2} {\sqrt{2}}^{2} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.2

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

$\int \sqrt{4 {\sqrt{2}}^{2} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\int \sqrt{4 {(2^{\frac{1}{2}})}^{2} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.3.3

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

$\int \sqrt{4 \cdot 2^{\frac{2}{2}} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.1.3.4.1

Cancel the common factor.

$\int \sqrt{4 \cdot 2^{\frac{2}{2}} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.3.4.2

Rewrite the expression.

$\int \sqrt{4 \cdot 2^{1} - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.3.5

Evaluate the exponent.

$\int \sqrt{4 \cdot 2 - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.4

Multiply $4$ by $2$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.5

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6

Combine and simplify the denominator.

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Step 2.1.1.6.1

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.4

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

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.5

Add $1$ and $1$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.6.3

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

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.6.6.4.1

Cancel the common factor.

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.6.4.2

Rewrite the expression.

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{3^{1}} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.6.6.5

Evaluate the exponent.

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{2} \sqrt{3}}{3} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.7

Simplify the numerator.

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Step 2.1.1.7.1

Combine using the product rule for radicals.

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{3 \cdot 2}}{3} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.7.2

Multiply $3$ by $2$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{6}}{3} \sin (t))}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.8

Combine $\frac{2 \sqrt{6}}{3}$ and $\sin (t)$ .

$\int \sqrt{8 - 3 {(\frac{2 \sqrt{6} \sin (t)}{3})}^{2}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.9

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

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Step 2.1.1.9.1

Apply the product rule to $\frac{2 \sqrt{6} \sin (t)}{3}$ .

$\int \sqrt{8 - 3 \frac{{(2 \sqrt{6} \sin (t))}^{2}}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.9.2

Apply the product rule to $2 \sqrt{6} \sin (t)$ .

$\int \sqrt{8 - 3 \frac{{(2 \sqrt{6})}^{2} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.9.3

Apply the product rule to $2 \sqrt{6}$ .

$\int \sqrt{8 - 3 \frac{2^{2} {\sqrt{6}}^{2} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10

Simplify the numerator.

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Step 2.1.1.10.1

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

$\int \sqrt{8 - 3 \frac{4 {\sqrt{6}}^{2} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.1.10.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\int \sqrt{8 - 3 \frac{4 {(6^{\frac{1}{2}})}^{2} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sqrt{8 - 3 \frac{4 \cdot 6^{\frac{1}{2} \cdot 2} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10.2.3

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

$\int \sqrt{8 - 3 \frac{4 \cdot 6^{\frac{2}{2}} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10.2.4

Cancel the common factor of $2$ .

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Step 2.1.1.10.2.4.1

Cancel the common factor.

$\int \sqrt{8 - 3 \frac{4 \cdot 6^{\frac{2}{2}} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10.2.4.2

Rewrite the expression.

$\int \sqrt{8 - 3 \frac{4 \cdot 6^{1} \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10.2.5

Evaluate the exponent.

$\int \sqrt{8 - 3 \frac{4 \cdot 6 \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.10.3

Multiply $4$ by $6$ .

$\int \sqrt{8 - 3 \frac{24 \sin^{2} (t)}{3^{2}}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.11

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

$\int \sqrt{8 - 3 \frac{24 \sin^{2} (t)}{9}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.12

Cancel the common factor of $3$ .

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Step 2.1.1.12.1

Factor $3$ out of $- 3$ .

$\int \sqrt{8 + 3 (- 1) \frac{24 \sin^{2} (t)}{9}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.12.2

Factor $3$ out of $9$ .

$\int \sqrt{8 + 3 \cdot - 1 \frac{24 \sin^{2} (t)}{3 \cdot 3}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.12.3

Cancel the common factor.

$\int \sqrt{8 + 3 \cdot - 1 \frac{24 \sin^{2} (t)}{3 \cdot 3}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.12.4

Rewrite the expression.

$\int \sqrt{8 - 1 \frac{24 \sin^{2} (t)}{3}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.13

Cancel the common factor of $24$ and $3$ .

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Step 2.1.1.13.1

Factor $3$ out of $24 \sin^{2} (t)$ .

$\int \sqrt{8 - 1 \frac{3 (8 \sin^{2} (t))}{3}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.13.2

Cancel the common factors.

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Step 2.1.1.13.2.1

Factor $3$ out of $3$ .

$\int \sqrt{8 - 1 \frac{3 (8 \sin^{2} (t))}{3 (1)}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.13.2.2

Cancel the common factor.

$\int \sqrt{8 - 1 \frac{3 (8 \sin^{2} (t))}{3 \cdot 1}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.13.2.3

Rewrite the expression.

$\int \sqrt{8 - 1 \frac{8 \sin^{2} (t)}{1}} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.13.2.4

Divide $8 \sin^{2} (t)$ by $1$ .

$\int \sqrt{8 - 1 (8 \sin^{2} (t))} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.1.14

Multiply $8$ by $- 1$ .

$\int \sqrt{8 - 8 \sin^{2} (t)} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.2

Factor $8$ out of $8$ .

$\int \sqrt{8 (1) - 8 \sin^{2} (t)} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.3

Factor $8$ out of $- 8 \sin^{2} (t)$ .

$\int \sqrt{8 (1) + 8 (- \sin^{2} (t))} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.4

Factor $8$ out of $8 (1) + 8 (- \sin^{2} (t))$ .

$\int \sqrt{8 (1 - \sin^{2} (t))} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.5

Apply pythagorean identity.

$\int \sqrt{8 \cos^{2} (t)} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.6

Rewrite $8 \cos^{2} (t)$ as ${(2 \cos (t))}^{2} \cdot 2$ .

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Step 2.1.6.1

Factor $4$ out of $8$ .

$\int \sqrt{4 (2) \cos^{2} (t)} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.6.2

Rewrite $4$ as $2^{2}$ .

$\int \sqrt{2^{2} \cdot 2 \cos^{2} (t)} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.6.3

Move $2$ .

$\int \sqrt{2^{2} \cos^{2} (t) \cdot 2} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.6.4

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

$\int \sqrt{{(2 \cos (t))}^{2} \cdot 2} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.1.7

Pull terms out from under the radical.

$\int 2 \cos (t) \sqrt{2} \frac{2 \sqrt{6} \cos (t)}{3} d t$

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{2 \sqrt{6} \cos (t)}{3}$ and $2$ .

$\int \frac{2 \sqrt{6} \cos (t) \cdot 2}{3} \cos (t) \sqrt{2} d t$

Step 2.2.2

Multiply $2$ by $2$ .

$\int \frac{4 \sqrt{6} \cos (t)}{3} \cos (t) \sqrt{2} d t$

Step 2.2.3

Combine $\frac{4 \sqrt{6} \cos (t)}{3}$ and $\cos (t)$ .

$\int \frac{4 \sqrt{6} \cos (t) \cos (t)}{3} \sqrt{2} d t$

Step 2.2.4

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

$\int \frac{4 \sqrt{6} (\cos^{1} (t) \cos (t))}{3} \sqrt{2} d t$

Step 2.2.5

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

$\int \frac{4 \sqrt{6} (\cos^{1} (t) \cos^{1} (t))}{3} \sqrt{2} d t$

Step 2.2.6

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

$\int \frac{4 \sqrt{6} {\cos (t)}^{1 + 1}}{3} \sqrt{2} d t$

Step 2.2.7

Add $1$ and $1$ .

$\int \frac{4 \sqrt{6} \cos^{2} (t)}{3} \sqrt{2} d t$

Step 2.2.8

Combine $\frac{4 \sqrt{6} \cos^{2} (t)}{3}$ and $\sqrt{2}$ .

$\int \frac{4 \sqrt{6} \cos^{2} (t) \sqrt{2}}{3} d t$

Step 2.2.9

Combine using the product rule for radicals.

$\int \frac{4 \sqrt{2 \cdot 6} \cos^{2} (t)}{3} d t$

Step 2.2.10

Multiply $2$ by $6$ .

$\int \frac{4 \sqrt{12} \cos^{2} (t)}{3} d t$

Step 3

Since $\frac{4 \sqrt{12}}{3}$ is constant with respect to $t$ , move $\frac{4 \sqrt{12}}{3}$ out of the integral.

$\frac{4 \sqrt{12}}{3} \int \cos^{2} (t) d t$

Step 4

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

$\frac{4 \sqrt{12}}{3} \int \frac{1 + \cos (2 t)}{2} d t$

Step 5

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

$\frac{4 \sqrt{12}}{3} (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1}{2}$ by $\frac{4 \sqrt{12}}{3}$ .

$\frac{4 \sqrt{12}}{2 \cdot 3} \int 1 + \cos (2 t) d t$

Step 6.2

Multiply $2$ by $3$ .

$\frac{4 \sqrt{12}}{6} \int 1 + \cos (2 t) d t$

Step 6.3

Cancel the common factor of $4$ and $6$ .

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Step 6.3.1

Factor $2$ out of $4 \sqrt{12}$ .

$\frac{2 (2 \sqrt{12})}{6} \int 1 + \cos (2 t) d t$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $6$ .

$\frac{2 (2 \sqrt{12})}{2 (3)} \int 1 + \cos (2 t) d t$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{12})}{2 \cdot 3} \int 1 + \cos (2 t) d t$

Step 6.3.2.3

Rewrite the expression.

$\frac{2 \sqrt{12}}{3} \int 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$\frac{2 \sqrt{12}}{3} (\int d t + \int \cos (2 t) d t)$

Step 8

Apply the constant rule.

$\frac{2 \sqrt{12}}{3} (t + C + \int \cos (2 t) d t)$

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 $d u$ .

$\frac{2 \sqrt{12}}{3} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 10

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

$\frac{2 \sqrt{12}}{3} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 11

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

$\frac{2 \sqrt{12}}{3} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 12

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

$\frac{2 \sqrt{12}}{3} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 13

Simplify.

$\frac{4 \sqrt{3}}{3} (t + \frac{1}{2} \sin (u)) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $t$ with $\arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}})$ .

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

Step 14.2

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

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

Step 14.3

Replace all occurrences of $t$ with $\arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}})$ .

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

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $\frac{\sqrt{3} x}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 15.1.2

Combine and simplify the denominator.

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Step 15.1.2.1

Multiply $\frac{\sqrt{3} x}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 15.1.2.2

Move $\sqrt{2}$ .

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

Step 15.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 15.1.2.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 15.1.2.5

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

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

Step 15.1.2.6

Add $1$ and $1$ .

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

Step 15.1.2.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.1.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 15.1.2.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 15.1.2.7.3

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

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

Step 15.1.2.7.4

Cancel the common factor of $2$ .

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Step 15.1.2.7.4.1

Cancel the common factor.

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

Step 15.1.2.7.4.2

Rewrite the expression.

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

Step 15.1.2.7.5

Evaluate the exponent.

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

Step 15.1.3

Simplify the numerator.

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Step 15.1.3.1

Combine using the product rule for radicals.

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

Step 15.1.3.2

Multiply $3$ by $2$ .

$\frac{4 \sqrt{3}}{3} (\arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{6}}{2 \cdot 2}))) + C$

Step 15.1.4

Multiply $2$ by $2$ .

$\frac{4 \sqrt{3}}{3} (\arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))) + C$

Step 15.1.5

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (\frac{x \sqrt{6}}{4}))$ .

$\frac{4 \sqrt{3}}{3} (\arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}}) + \frac{\sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{2}) + C$

Step 15.2

Apply the distributive property.

$\frac{4 \sqrt{3}}{3} \arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}}) + \frac{4 \sqrt{3}}{3} \cdot \frac{\sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{2} + C$

Step 15.3

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

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}})}{3} + \frac{4 \sqrt{3}}{3} \cdot \frac{\sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{2} + C$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Factor $2$ out of $4 \sqrt{3}$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}})}{3} + \frac{2 (2 \sqrt{3})}{3} \cdot \frac{\sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{2} + C$

Step 15.4.2

Cancel the common factor.

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}})}{3} + \frac{2 (2 \sqrt{3})}{3} \cdot \frac{\sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{2} + C$

Step 15.4.3

Rewrite the expression.

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}})}{3} + \frac{2 \sqrt{3}}{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4})) + C$

Step 15.5

Combine $\frac{2 \sqrt{3}}{3}$ and $\sin (2 \arcsin (\frac{x \sqrt{6}}{4}))$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16

Simplify.

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Step 16.1

Multiply $\frac{\sqrt{3} x}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2

Combine and simplify the denominator.

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Step 16.2.1

Multiply $\frac{\sqrt{3} x}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 \sqrt{2} \sqrt{2}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.2

Move $\sqrt{2}$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 (\sqrt{2} \sqrt{2})})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.5

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

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.6

Add $1$ and $1$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 {\sqrt{2}}^{2}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 16.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.7.3

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

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.7.4

Cancel the common factor of $2$ .

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Step 16.2.7.4.1

Cancel the common factor.

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.7.4.2

Rewrite the expression.

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 \cdot 2^{1}})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.2.7.5

Evaluate the exponent.

$\frac{4 \sqrt{3} \arcsin (\frac{\sqrt{3} x \sqrt{2}}{2 \cdot 2})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.3

Simplify the numerator.

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Step 16.3.1

Combine using the product rule for radicals.

$\frac{4 \sqrt{3} \arcsin (\frac{x \sqrt{3 \cdot 2}}{2 \cdot 2})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.3.2

Multiply $3$ by $2$ .

$\frac{4 \sqrt{3} \arcsin (\frac{x \sqrt{6}}{2 \cdot 2})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.4

Multiply $2$ by $2$ .

$\frac{4 \sqrt{3} \arcsin (\frac{x \sqrt{6}}{4})}{3} + \frac{2 \sqrt{3} \sin (2 \arcsin (\frac{x \sqrt{6}}{4}))}{3} + C$

Step 16.5

Reorder terms.

$\frac{4 \sqrt{3}}{3} \arcsin (\frac{\sqrt{6}}{4} x) + \frac{2 \sqrt{3}}{3} \sin (2 \arcsin (\frac{\sqrt{6}}{4} x)) + C$