Calculus Examples

$\int_{- \frac{π}{3}}^{\frac{π}{3}} 4 π (1 - \cos (x)) d x$

Step 1

Since $4 π$ is constant with respect to $x$ , move $4 π$ out of the integral.

$4 π \int_{- \frac{π}{3}}^{\frac{π}{3}} 1 - \cos (x) d x$

Step 2

Split the single integral into multiple integrals.

$4 π (\int_{- \frac{π}{3}}^{\frac{π}{3}} d x + \int_{- \frac{π}{3}}^{\frac{π}{3}} - \cos (x) d x)$

Step 3

Apply the constant rule.

$4 π (x]_{- \frac{π}{3}}^{\frac{π}{3}} + \int_{- \frac{π}{3}}^{\frac{π}{3}} - \cos (x) d x)$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$4 π (x]_{- \frac{π}{3}}^{\frac{π}{3}} - \int_{- \frac{π}{3}}^{\frac{π}{3}} \cos (x) d x)$

Step 5

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

$4 π (x]_{- \frac{π}{3}}^{\frac{π}{3}} - (\sin (x)]_{- \frac{π}{3}}^{\frac{π}{3}}))$

Step 6

Simplify the answer.

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Step 6.1

Substitute and simplify.

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Step 6.1.1

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

$4 π ((\frac{π}{3}) + \frac{π}{3} - (\sin (x)]_{- \frac{π}{3}}^{\frac{π}{3}}))$

Step 6.1.2

Evaluate $\sin (x)$ at $\frac{π}{3}$ and at $- \frac{π}{3}$ .

$4 π (\frac{π}{3} + \frac{π}{3} - (\sin (\frac{π}{3}) - \sin (- \frac{π}{3})))$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Combine the numerators over the common denominator.

$4 π (\frac{π + π}{3} - (\sin (\frac{π}{3}) - \sin (- \frac{π}{3})))$

Step 6.1.3.2

Add $π$ and $π$ .

$4 π (\frac{2 π}{3} - (\sin (\frac{π}{3}) - \sin (- \frac{π}{3})))$

Step 6.2

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

$4 π (\frac{2 π}{3} - (\frac{\sqrt{3}}{2} - \sin (- \frac{π}{3})))$

Step 6.3

Simplify.

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Step 6.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$4 π (\frac{2 π}{3} - (\frac{\sqrt{3}}{2} - \sin (\frac{5 π}{3})))$

Step 6.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$4 π (\frac{2 π}{3} - (\frac{\sqrt{3}}{2} - - \sin (\frac{π}{3})))$

Step 6.3.3

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

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

Step 6.3.4

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

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Step 6.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.3.4.2

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

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

Step 6.3.5

Combine the numerators over the common denominator.

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

Step 6.3.6

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

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

Step 6.3.7

Cancel the common factor of $2$ .

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Step 6.3.7.1

Cancel the common factor.

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

Step 6.3.7.2

Divide $\sqrt{3}$ by $1$ .

$4 π (\frac{2 π}{3} - \sqrt{3})$

Step 6.3.8

Apply the distributive property.

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

Step 6.3.9

Multiply $4 π \frac{2 π}{3}$ .

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Step 6.3.9.1

Combine $\frac{2 π}{3}$ and $4$ .

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

Step 6.3.9.2

Multiply $4$ by $2$ .

$\frac{8 π}{3} π + 4 π (- \sqrt{3})$

Step 6.3.9.3

Combine $\frac{8 π}{3}$ and $π$ .

$\frac{8 π π}{3} + 4 π (- \sqrt{3})$

Step 6.3.9.4

Raise $π$ to the power of $1$ .

$\frac{8 (π^{1} π)}{3} + 4 π (- \sqrt{3})$

Step 6.3.9.5

Raise $π$ to the power of $1$ .

$\frac{8 (π^{1} π^{1})}{3} + 4 π (- \sqrt{3})$

Step 6.3.9.6

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

$\frac{8 π^{1 + 1}}{3} + 4 π (- \sqrt{3})$

Step 6.3.9.7

Add $1$ and $1$ .

$\frac{8 π^{2}}{3} + 4 π (- \sqrt{3})$

Step 6.3.10

Multiply $- 1$ by $4$ .

$\frac{8 π^{2}}{3} - 4 π \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{8 π^{2}}{3} - 4 π \sqrt{3}$

Decimal Form:

$4.55335269 \dots$