Calculus Examples

$\int_{0}^{\frac{π}{3}} {(\cos (x) + \sec (x))}^{2} d x$

Step 1

Simplify.

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Step 1.1

Rewrite ${(\cos (x) + \sec (x))}^{2}$ as $(\cos (x) + \sec (x)) (\cos (x) + \sec (x))$ .

$\int_{0}^{\frac{π}{3}} (\cos (x) + \sec (x)) (\cos (x) + \sec (x)) d x$

Step 1.2

Expand $(\cos (x) + \sec (x)) (\cos (x) + \sec (x))$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\int_{0}^{\frac{π}{3}} \cos (x) (\cos (x) + \sec (x)) + \sec (x) (\cos (x) + \sec (x)) d x$

Step 1.2.2

Apply the distributive property.

$\int_{0}^{\frac{π}{3}} \cos (x) \cos (x) + \cos (x) \sec (x) + \sec (x) (\cos (x) + \sec (x)) d x$

Step 1.2.3

Apply the distributive property.

$\int_{0}^{\frac{π}{3}} \cos (x) \cos (x) + \cos (x) \sec (x) + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $\cos (x) \cos (x)$ .

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Step 1.3.1.1.1

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

$\int_{0}^{\frac{π}{3}} \cos^{1} (x) \cos (x) + \cos (x) \sec (x) + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.1.2

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

$\int_{0}^{\frac{π}{3}} \cos^{1} (x) \cos^{1} (x) + \cos (x) \sec (x) + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.1.3

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

$\int_{0}^{\frac{π}{3}} {\cos (x)}^{1 + 1} + \cos (x) \sec (x) + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.1.4

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + \cos (x) \sec (x) + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.1.2.1

Reorder $\cos (x)$ and $\sec (x)$ .

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + \sec (x) \cos (x) + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.2.2

Rewrite $\cos (x) \sec (x)$ in terms of sines and cosines.

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + \frac{1}{\cos (x)} \cos (x) + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.2.3

Cancel the common factors.

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 1 + \sec (x) \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.1.3.1

Rewrite $\sec (x) \cos (x)$ in terms of sines and cosines.

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 1 + \frac{1}{\cos (x)} \cos (x) + \sec (x) \sec (x) d x$

Step 1.3.1.3.2

Cancel the common factors.

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 1 + 1 + \sec (x) \sec (x) d x$

Step 1.3.1.4

Multiply $\sec (x) \sec (x)$ .

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Step 1.3.1.4.1

Raise $\sec (x)$ to the power of $1$ .

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 1 + 1 + \sec^{1} (x) \sec (x) d x$

Step 1.3.1.4.2

Raise $\sec (x)$ to the power of $1$ .

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 1 + 1 + \sec^{1} (x) \sec^{1} (x) d x$

Step 1.3.1.4.3

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

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 1 + 1 + {\sec (x)}^{1 + 1} d x$

Step 1.3.1.4.4

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 1 + 1 + \sec^{2} (x) d x$

Step 1.3.2

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) + 2 + \sec^{2} (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{\frac{π}{3}} \cos^{2} (x) d x + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 3

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

$\int_{0}^{\frac{π}{3}} \frac{1 + \cos (2 x)}{2} d x + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 4

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

$\frac{1}{2} \int_{0}^{\frac{π}{3}} 1 + \cos (2 x) d x + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 5

Split the single integral into multiple integrals.

$\frac{1}{2} (\int_{0}^{\frac{π}{3}} d x + \int_{0}^{\frac{π}{3}} \cos (2 x) d x) + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 6

Apply the constant rule.

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \int_{0}^{\frac{π}{3}} \cos (2 x) d x) + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $2 x$ .

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

Step 7.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 7.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 7.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \frac{π}{3}$

Step 7.5

Combine $2$ and $\frac{π}{3}$ .

$u_{upper} = \frac{2 π}{3}$

Step 7.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \frac{2 π}{3}$

Step 7.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \int_{0}^{\frac{2 π}{3}} \cos (u) \frac{1}{2} d u) + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 8

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

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \int_{0}^{\frac{2 π}{3}} \frac{\cos (u)}{2} d u) + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 9

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

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \frac{1}{2} \int_{0}^{\frac{2 π}{3}} \cos (u) d u) + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 10

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

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \frac{1}{2} \sin (u)]_{0}^{\frac{2 π}{3}}) + \int_{0}^{\frac{π}{3}} 2 d x + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 11

Apply the constant rule.

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \frac{1}{2} \sin (u)]_{0}^{\frac{2 π}{3}}) + 2 x]_{0}^{\frac{π}{3}} + \int_{0}^{\frac{π}{3}} \sec^{2} (x) d x$

Step 12

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \frac{1}{2} \sin (u)]_{0}^{\frac{2 π}{3}}) + 2 x]_{0}^{\frac{π}{3}} + \tan (x)]_{0}^{\frac{π}{3}}$

Step 13

Combine $2 x]_{0}^{\frac{π}{3}}$ and $\tan (x)]_{0}^{\frac{π}{3}}$ .

$\frac{1}{2} (x]_{0}^{\frac{π}{3}} + \frac{1}{2} \sin (u)]_{0}^{\frac{2 π}{3}}) + 2 x + \tan (x)]_{0}^{\frac{π}{3}}$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $x$ at $\frac{π}{3}$ and at $0$ .

$\frac{1}{2} ((\frac{π}{3}) + 0 + \frac{1}{2} \sin (u)]_{0}^{\frac{2 π}{3}}) + 2 x + \tan (x)]_{0}^{\frac{π}{3}}$

Step 14.2

Evaluate $\sin (u)$ at $\frac{2 π}{3}$ and at $0$ .

$\frac{1}{2} ((\frac{π}{3}) + 0 + \frac{1}{2} ((\sin (\frac{2 π}{3})) - \sin (0))) + 2 x + \tan (x)]_{0}^{\frac{π}{3}}$

Step 14.3

Evaluate $2 x + \tan (x)$ at $\frac{π}{3}$ and at $0$ .

$\frac{1}{2} ((\frac{π}{3}) + 0 + \frac{1}{2} ((\sin (\frac{2 π}{3})) - \sin (0))) + (2 \frac{π}{3} + \tan (\frac{π}{3})) - (2 \cdot 0 + \tan (0))$

Step 14.4

Simplify.

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Step 14.4.1

Add $\frac{π}{3}$ and $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} ((\sin (\frac{2 π}{3})) - \sin (0))) + (2 \frac{π}{3} + \tan (\frac{π}{3})) - (2 \cdot 0 + \tan (0))$

Step 14.4.2

Combine $2$ and $\frac{π}{3}$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} (\sin (\frac{2 π}{3}) - \sin (0))) + \frac{2 π}{3} + \tan (\frac{π}{3}) - (2 \cdot 0 + \tan (0))$

Step 14.4.3

Multiply $2$ by $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} (\sin (\frac{2 π}{3}) - \sin (0))) + \frac{2 π}{3} + \tan (\frac{π}{3}) - (0 + \tan (0))$

Step 14.4.4

Add $0$ and $\tan (0)$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} (\sin (\frac{2 π}{3}) - \sin (0))) + \frac{2 π}{3} + \tan (\frac{π}{3}) - \tan (0)$

Step 15

Simplify.

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Step 15.1

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} (\sin (\frac{2 π}{3}) - 0)) + \frac{2 π}{3} + \tan (\frac{π}{3}) - \tan (0)$

Step 15.2

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} (\sin (\frac{2 π}{3}) - 0)) + \frac{2 π}{3} + \sqrt{3} - \tan (0)$

Step 15.3

The exact value of $\tan (0)$ is $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} (\sin (\frac{2 π}{3}) - 0)) + \frac{2 π}{3} + \sqrt{3} - 0$

Step 15.4

Multiply $- 1$ by $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} (\sin (\frac{2 π}{3}) + 0)) + \frac{2 π}{3} + \sqrt{3} - 0$

Step 15.5

Add $\sin (\frac{2 π}{3})$ and $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{1}{2} \sin (\frac{2 π}{3})) + \frac{2 π}{3} + \sqrt{3} - 0$

Step 15.6

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

$\frac{1}{2} (\frac{π}{3} + \frac{\sin (\frac{2 π}{3})}{2}) + \frac{2 π}{3} + \sqrt{3} - 0$

Step 15.7

Multiply $- 1$ by $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{\sin (\frac{2 π}{3})}{2}) + \frac{2 π}{3} + \sqrt{3} + 0$

Step 15.8

Add $\frac{1}{2} (\frac{π}{3} + \frac{\sin (\frac{2 π}{3})}{2}) + \frac{2 π}{3} + \sqrt{3}$ and $0$ .

$\frac{1}{2} (\frac{π}{3} + \frac{\sin (\frac{2 π}{3})}{2}) + \frac{2 π}{3} + \sqrt{3}$

Step 16

Simplify.

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Step 16.1

Simplify each term.

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Step 16.1.1

Simplify the numerator.

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Step 16.1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{1}{2} (\frac{π}{3} + \frac{\sin (\frac{π}{3})}{2}) + \frac{2 π}{3} + \sqrt{3}$

Step 16.1.1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} (\frac{π}{3} + \frac{\frac{\sqrt{3}}{2}}{2}) + \frac{2 π}{3} + \sqrt{3}$

Step 16.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} (\frac{π}{3} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2}) + \frac{2 π}{3} + \sqrt{3}$

Step 16.1.3

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{1}{2}$ .

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Step 16.1.3.1

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

$\frac{1}{2} (\frac{π}{3} + \frac{\sqrt{3}}{2 \cdot 2}) + \frac{2 π}{3} + \sqrt{3}$

Step 16.1.3.2

Multiply $2$ by $2$ .

$\frac{1}{2} (\frac{π}{3} + \frac{\sqrt{3}}{4}) + \frac{2 π}{3} + \sqrt{3}$

Step 16.2

Apply the distributive property.

$\frac{1}{2} \cdot \frac{π}{3} + \frac{1}{2} \cdot \frac{\sqrt{3}}{4} + \frac{2 π}{3} + \sqrt{3}$

Step 16.3

Multiply $\frac{1}{2} \cdot \frac{π}{3}$ .

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Step 16.3.1

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

$\frac{π}{2 \cdot 3} + \frac{1}{2} \cdot \frac{\sqrt{3}}{4} + \frac{2 π}{3} + \sqrt{3}$

Step 16.3.2

Multiply $2$ by $3$ .

$\frac{π}{6} + \frac{1}{2} \cdot \frac{\sqrt{3}}{4} + \frac{2 π}{3} + \sqrt{3}$

Step 16.4

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

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Step 16.4.1

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

$\frac{π}{6} + \frac{\sqrt{3}}{2 \cdot 4} + \frac{2 π}{3} + \sqrt{3}$

Step 16.4.2

Multiply $2$ by $4$ .

$\frac{π}{6} + \frac{\sqrt{3}}{8} + \frac{2 π}{3} + \sqrt{3}$

Step 16.5

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{π}{6} + \frac{2 π}{3} \cdot \frac{2}{2} + \frac{\sqrt{3}}{8} + \sqrt{3}$

Step 16.6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.6.1

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

$\frac{π}{6} + \frac{2 π \cdot 2}{3 \cdot 2} + \frac{\sqrt{3}}{8} + \sqrt{3}$

Step 16.6.2

Multiply $3$ by $2$ .

$\frac{π}{6} + \frac{2 π \cdot 2}{6} + \frac{\sqrt{3}}{8} + \sqrt{3}$

Step 16.7

Combine the numerators over the common denominator.

$\frac{π + 2 π \cdot 2}{6} + \frac{\sqrt{3}}{8} + \sqrt{3}$

Step 16.8

To write $\frac{π + 2 π \cdot 2}{6}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π + 2 π \cdot 2}{6} \cdot \frac{4}{4} + \frac{\sqrt{3}}{8} + \sqrt{3}$

Step 16.9

To write $\frac{\sqrt{3}}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π + 2 π \cdot 2}{6} \cdot \frac{4}{4} + \frac{\sqrt{3}}{8} \cdot \frac{3}{3} + \sqrt{3}$

Step 16.10

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.10.1

Multiply $\frac{π + 2 π \cdot 2}{6}$ by $\frac{4}{4}$ .

$\frac{(π + 2 π \cdot 2) \cdot 4}{6 \cdot 4} + \frac{\sqrt{3}}{8} \cdot \frac{3}{3} + \sqrt{3}$

Step 16.10.2

Multiply $6$ by $4$ .

$\frac{(π + 2 π \cdot 2) \cdot 4}{24} + \frac{\sqrt{3}}{8} \cdot \frac{3}{3} + \sqrt{3}$

Step 16.10.3

Multiply $\frac{\sqrt{3}}{8}$ by $\frac{3}{3}$ .

$\frac{(π + 2 π \cdot 2) \cdot 4}{24} + \frac{\sqrt{3} \cdot 3}{8 \cdot 3} + \sqrt{3}$

Step 16.10.4

Multiply $8$ by $3$ .

$\frac{(π + 2 π \cdot 2) \cdot 4}{24} + \frac{\sqrt{3} \cdot 3}{24} + \sqrt{3}$

Step 16.11

Combine the numerators over the common denominator.

$\frac{(π + 2 π \cdot 2) \cdot 4 + \sqrt{3} \cdot 3}{24} + \sqrt{3}$

Step 16.12

Reorder terms.

$\frac{4 (π + 2 π \cdot 2) + 3 \sqrt{3}}{24} + \sqrt{3}$

Step 16.13

Combine $\frac{4 (π + 2 π \cdot 2) + 3 \sqrt{3}}{24}$ and $\sqrt{3}$ using a common denominator.

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Step 16.13.1

Move $π$ .

$\frac{4 (π + 2 \cdot 2 π) + 3 \sqrt{3}}{24} + \sqrt{3}$

Step 16.13.2

To write $\sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$\frac{4 (π + 2 \cdot 2 π) + 3 \sqrt{3}}{24} + \sqrt{3} \cdot \frac{24}{24}$

Step 16.13.3

Combine $\sqrt{3}$ and $\frac{24}{24}$ .

$\frac{4 (π + 2 \cdot 2 π) + 3 \sqrt{3}}{24} + \frac{\sqrt{3} \cdot 24}{24}$

Step 16.13.4

Combine the numerators over the common denominator.

$\frac{4 (π + 2 \cdot 2 π) + 3 \sqrt{3} + \sqrt{3} \cdot 24}{24}$

Step 16.14

Reorder the factors of $\sqrt{3} \cdot 24$ .

$\frac{4 (π + 2 \cdot 2 π) + 3 \sqrt{3} + 24 \sqrt{3}}{24}$

Step 16.15

Add $3 \sqrt{3}$ and $24 \sqrt{3}$ .

$\frac{4 (π + 2 \cdot 2 π) + 27 \sqrt{3}}{24}$

Step 16.16

Multiply $2$ by $2$ .

$\frac{4 (π + 4 π) + 27 \sqrt{3}}{24}$

Step 16.17

Add $π$ and $4 π$ .

$\frac{4 \cdot 5 π + 27 \sqrt{3}}{24}$

Step 17

Multiply $4$ by $5$ .

$\frac{20 π + 27 \sqrt{3}}{24}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{20 π + 27 \sqrt{3}}{24}$

Decimal Form:

$4.56655103 \dots$