Calculus Examples

$\int_{0}^{\frac{π}{6}} {(3 \cos (3 θ))}^{2} d θ$

Step 1

Simplify.

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Step 1.1

Factor $3$ out of $3 \cos (3 θ)$ .

$\int_{0}^{\frac{π}{6}} {(3 (\cos (3 θ)))}^{2} d θ$

Step 1.2

Apply the product rule to $3 (\cos (3 θ))$ .

$\int_{0}^{\frac{π}{6}} 3^{2} \cos^{2} (3 θ) d θ$

Step 1.3

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

$\int_{0}^{\frac{π}{6}} 9 \cos^{2} (3 θ) d θ$

Step 2

Since $9$ is constant with respect to $θ$ , move $9$ out of the integral.

$9 \int_{0}^{\frac{π}{6}} \cos^{2} (3 θ) d θ$

Step 3

Let $u_{1} = 3 θ$ . Then $d u_{1} = 3 d θ$ , so $\frac{1}{3} d u_{1} = d θ$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = 3 θ$ . Find $\frac{d u_{1}}{d θ}$ .

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Step 3.1.1

Differentiate $3 θ$ .

$\frac{d}{d θ} [3 θ]$

Step 3.1.2

Since $3$ is constant with respect to $θ$ , the derivative of $3 θ$ with respect to $θ$ is $3 \frac{d}{d θ} [θ]$ .

$3 \frac{d}{d θ} [θ]$

Step 3.1.3

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

$3 \cdot 1$

Step 3.1.4

Multiply $3$ by $1$ .

$3$

Step 3.2

Substitute the lower limit in for $θ$ in $u_{1} = 3 θ$ .

$u_{lower} = 3 \cdot 0$

Step 3.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 3.4

Substitute the upper limit in for $θ$ in $u_{1} = 3 θ$ .

$u_{upper} = 3 \frac{π}{6}$

Step 3.5

Cancel the common factor of $3$ .

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Step 3.5.1

Factor $3$ out of $6$ .

$u_{upper} = 3 \frac{π}{3 (2)}$

Step 3.5.2

Cancel the common factor.

$u_{upper} = 3 \frac{π}{3 \cdot 2}$

Step 3.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 3.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 3.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$9 \int_{0}^{\frac{π}{2}} \cos^{2} (u_{1}) \frac{1}{3} d u_{1}$

Step 4

Combine $\cos^{2} (u_{1})$ and $\frac{1}{3}$ .

$9 \int_{0}^{\frac{π}{2}} \frac{\cos^{2} (u_{1})}{3} d u_{1}$

Step 5

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$9 (\frac{1}{3} \int_{0}^{\frac{π}{2}} \cos^{2} (u_{1}) d u_{1})$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{3}$ and $9$ .

$\frac{9}{3} \int_{0}^{\frac{π}{2}} \cos^{2} (u_{1}) d u_{1}$

Step 6.2

Cancel the common factor of $9$ and $3$ .

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Step 6.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 3}{3} \int_{0}^{\frac{π}{2}} \cos^{2} (u_{1}) d u_{1}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 3}{3 (1)} \int_{0}^{\frac{π}{2}} \cos^{2} (u_{1}) d u_{1}$

Step 6.2.2.2

Cancel the common factor.

$\frac{3 \cdot 3}{3 \cdot 1} \int_{0}^{\frac{π}{2}} \cos^{2} (u_{1}) d u_{1}$

Step 6.2.2.3

Rewrite the expression.

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

Step 6.2.2.4

Divide $3$ by $1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Combine $\frac{1}{2}$ and $3$ .

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

Step 10

Split the single integral into multiple integrals.

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

Step 11

Apply the constant rule.

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

Step 12

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

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Step 12.1

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 12.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 12.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 12.1.3

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

$2 \cdot 1$

Step 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{lower} = 2 \cdot 0$

Step 12.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 12.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{upper} = 2 \frac{π}{2}$

Step 12.5

Cancel the common factor of $2$ .

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Step 12.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 12.5.2

Rewrite the expression.

$u_{upper} = π$

Step 12.6

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

$u_{lower} = 0$

$u_{upper} = π$

Step 12.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

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

Step 13

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

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

Step 14

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

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

Step 15

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

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

Step 16

Substitute and simplify.

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Step 16.1

Evaluate $u_{1}$ at $\frac{π}{2}$ and at $0$ .

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

Step 16.2

Evaluate $\sin (u_{2})$ at $π$ and at $0$ .

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

Step 16.3

Add $\frac{π}{2}$ and $0$ .

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

Step 17

Simplify.

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Step 17.1

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

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

Step 17.2

Multiply $- 1$ by $0$ .

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

Step 17.3

Add $\sin (π)$ and $0$ .

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

Step 17.4

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

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

Step 18

Simplify.

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Step 18.1

Combine the numerators over the common denominator.

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

Step 18.2

Simplify each term.

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Step 18.2.1

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

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

Step 18.2.2

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

$\frac{3}{2} \cdot \frac{π + 0}{2}$

Step 18.3

Add $π$ and $0$ .

$\frac{3}{2} \cdot \frac{π}{2}$

Step 18.4

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

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Step 18.4.1

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

$\frac{3 π}{2 \cdot 2}$

Step 18.4.2

Multiply $2$ by $2$ .

$\frac{3 π}{4}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{3 π}{4}$

Decimal Form:

$2.35619449 \dots$