Calculus Examples

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

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $\frac{1 - \cos (2 x)}{2}$ by $\frac{1 + \cos (2 x)}{2}$ .

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

Step 4.2

Multiply $2$ by $2$ .

$3 \int_{0}^{\frac{π}{2}} \frac{(1 - \cos (2 x)) (1 + \cos (2 x))}{4} d x$

Step 5

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$3 (\frac{1}{4} \int_{0}^{\frac{π}{2}} (1 - \cos (2 x)) (1 + \cos (2 x)) d x)$

Step 6

Combine $\frac{1}{4}$ and $3$ .

$\frac{3}{4} \int_{0}^{\frac{π}{2}} (1 - \cos (2 x)) (1 + \cos (2 x)) d x$

Step 7

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

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Step 7.1

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

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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 $u_{1} = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 7.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 7.4

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

$u_{upper} = 2 \frac{π}{2}$

Step 7.5

Cancel the common factor of $2$ .

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Step 7.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 7.5.2

Rewrite the expression.

$u_{upper} = π$

Step 7.6

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

$u_{lower} = 0$

$u_{upper} = π$

Step 7.7

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

$\frac{3}{4} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) \frac{1}{2} d u_{1}$

Step 8

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

$\frac{3}{4} (\frac{1}{2} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1})$

Step 9

Simplify by multiplying through.

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Step 9.1

Simplify.

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Step 9.1.1

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

$\frac{3}{2 \cdot 4} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 9.1.2

Multiply $2$ by $4$ .

$\frac{3}{8} \int_{0}^{π} (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Step 9.2

Expand $(1 - \cos (u_{1})) (1 + \cos (u_{1}))$ .

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Step 9.2.1

Apply the distributive property.

$\frac{3}{8} \int_{0}^{π} 1 (1 + \cos (u_{1})) - \cos (u_{1}) (1 + \cos (u_{1})) d u_{1}$

Step 9.2.2

Apply the distributive property.

$\frac{3}{8} \int_{0}^{π} 1 \cdot 1 + 1 \cos (u_{1}) - \cos (u_{1}) (1 + \cos (u_{1})) d u_{1}$

Step 9.2.3

Apply the distributive property.

$\frac{3}{8} \int_{0}^{π} 1 \cdot 1 + 1 \cos (u_{1}) - \cos (u_{1}) \cdot 1 - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 9.2.4

Move $\cos (u_{1})$ .

$\frac{3}{8} \int_{0}^{π} 1 \cdot 1 + 1 \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 9.2.5

Multiply $1$ by $1$ .

$\frac{3}{8} \int_{0}^{π} 1 + 1 \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 9.2.6

Multiply $\cos (u_{1})$ by $1$ .

$\frac{3}{8} \int_{0}^{π} 1 + \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 9.2.7

Multiply $- 1$ by $1$ .

$\frac{3}{8} \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Step 9.2.8

Factor out negative.

$\frac{3}{8} \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos (u_{1}) \cos (u_{1})) d u_{1}$

Step 9.2.9

Raise $\cos (u_{1})$ to the power of $1$ .

$\frac{3}{8} \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos^{1} (u_{1}) \cos (u_{1})) d u_{1}$

Step 9.2.10

Raise $\cos (u_{1})$ to the power of $1$ .

$\frac{3}{8} \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos^{1} (u_{1}) \cos^{1} (u_{1})) d u_{1}$

Step 9.2.11

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

$\frac{3}{8} \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - {\cos (u_{1})}^{1 + 1} d u_{1}$

Step 9.2.12

Add $1$ and $1$ .

$\frac{3}{8} \int_{0}^{π} 1 + \cos (u_{1}) - \cos (u_{1}) - \cos^{2} (u_{1}) d u_{1}$

Step 9.2.13

Subtract $\cos (u_{1})$ from $\cos (u_{1})$ .

$\frac{3}{8} \int_{0}^{π} 1 + 0 - \cos^{2} (u_{1}) d u_{1}$

Step 9.2.14

Subtract $\cos^{2} (u_{1})$ from $0$ .

$\frac{3}{8} \int_{0}^{π} 1 - \cos^{2} (u_{1}) d u_{1}$

Step 10

Split the single integral into multiple integrals.

$\frac{3}{8} (\int_{0}^{π} d u_{1} + \int_{0}^{π} - \cos^{2} (u_{1}) d u_{1})$

Step 11

Apply the constant rule.

$\frac{3}{8} (u_{1}]_{0}^{π} + \int_{0}^{π} - \cos^{2} (u_{1}) d u_{1})$

Step 12

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

$\frac{3}{8} (u_{1}]_{0}^{π} - \int_{0}^{π} \cos^{2} (u_{1}) d u_{1})$

Step 13

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

$\frac{3}{8} (u_{1}]_{0}^{π} - \int_{0}^{π} \frac{1 + \cos (2 u_{1})}{2} d u_{1})$

Step 14

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

$\frac{3}{8} (u_{1}]_{0}^{π} - (\frac{1}{2} \int_{0}^{π} 1 + \cos (2 u_{1}) d u_{1}))$

Step 15

Split the single integral into multiple integrals.

$\frac{3}{8} (u_{1}]_{0}^{π} - \frac{1}{2} (\int_{0}^{π} d u_{1} + \int_{0}^{π} \cos (2 u_{1}) d u_{1}))$

Step 16

Apply the constant rule.

$\frac{3}{8} (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \int_{0}^{π} \cos (2 u_{1}) d u_{1}))$

Step 17

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 17.1

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

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Step 17.1.1

Differentiate $2 u_{1}$ .

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

Step 17.1.2

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 17.1.3

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 17.1.4

Multiply $2$ by $1$ .

$2$

Step 17.2

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

$u_{lower} = 2 \cdot 0$

Step 17.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 17.4

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

$u_{upper} = 2 π$

Step 17.5

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

$u_{lower} = 0$

$u_{upper} = 2 π$

Step 17.6

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

$\frac{3}{8} (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \int_{0}^{2 π} \cos (u_{2}) \frac{1}{2} d u_{2}))$

Step 18

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

$\frac{3}{8} (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \int_{0}^{2 π} \frac{\cos (u_{2})}{2} d u_{2}))$

Step 19

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

$\frac{3}{8} (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \frac{1}{2} \int_{0}^{2 π} \cos (u_{2}) d u_{2}))$

Step 20

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

$\frac{3}{8} (u_{1}]_{0}^{π} - \frac{1}{2} (u_{1}]_{0}^{π} + \frac{1}{2} \sin (u_{2})]_{0}^{2 π}))$

Step 21

Substitute and simplify.

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Step 21.1

Evaluate $u_{1}$ at $π$ and at $0$ .

$\frac{3}{8} ((π) + 0 - \frac{1}{2} (u_{1}]_{0}^{π} + \frac{1}{2} \sin (u_{2})]_{0}^{2 π}))$

Step 21.2

Evaluate $u_{1}$ at $π$ and at $0$ .

$\frac{3}{8} (π + 0 - \frac{1}{2} (π + 0 + \frac{1}{2} \sin (u_{2})]_{0}^{2 π}))$

Step 21.3

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

$\frac{3}{8} (π + 0 - \frac{1}{2} (π + 0 + \frac{1}{2} ((\sin (2 π)) - \sin (0))))$

Step 21.4

Simplify.

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Step 21.4.1

Add $π$ and $0$ .

$\frac{3}{8} (π - \frac{1}{2} (π + 0 + \frac{1}{2} ((\sin (2 π)) - \sin (0))))$

Step 21.4.2

Add $π$ and $0$ .

$\frac{3}{8} (π - \frac{1}{2} (π + \frac{1}{2} (\sin (2 π) - \sin (0))))$

Step 22

Simplify.

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Step 22.1

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

$\frac{3}{8} (π - \frac{1}{2} (π + \frac{1}{2} (\sin (2 π) - 0)))$

Step 22.2

Multiply $- 1$ by $0$ .

$\frac{3}{8} (π - \frac{1}{2} (π + \frac{1}{2} (\sin (2 π) + 0)))$

Step 22.3

Add $\sin (2 π)$ and $0$ .

$\frac{3}{8} (π - \frac{1}{2} (π + \frac{1}{2} \sin (2 π)))$

Step 22.4

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

$\frac{3}{8} (π - \frac{1}{2} (π + \frac{\sin (2 π)}{2}))$

Step 23

Simplify.

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Step 23.1

Simplify each term.

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Step 23.1.1

Simplify the numerator.

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Step 23.1.1.1

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

$\frac{3}{8} (π - \frac{1}{2} (π + \frac{\sin (0)}{2}))$

Step 23.1.1.2

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

$\frac{3}{8} (π - \frac{1}{2} (π + \frac{0}{2}))$

Step 23.1.2

Divide $0$ by $2$ .

$\frac{3}{8} (π - \frac{1}{2} (π + 0))$

Step 23.2

Add $π$ and $0$ .

$\frac{3}{8} (π - \frac{1}{2} π)$

Step 23.3

Combine $π$ and $\frac{1}{2}$ .

$\frac{3}{8} (π - \frac{π}{2})$

Step 23.4

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

$\frac{3}{8} (π \cdot \frac{2}{2} - \frac{π}{2})$

Step 23.5

Combine $π$ and $\frac{2}{2}$ .

$\frac{3}{8} (\frac{π \cdot 2}{2} - \frac{π}{2})$

Step 23.6

Combine the numerators over the common denominator.

$\frac{3}{8} \cdot \frac{π \cdot 2 - π}{2}$

Step 23.7

Move $2$ to the left of $π$ .

$\frac{3}{8} \cdot \frac{2 \cdot π - π}{2}$

Step 23.8

Subtract $π$ from $2 π$ .

$\frac{3}{8} \cdot \frac{π}{2}$

Step 23.9

Multiply $\frac{3}{8} \cdot \frac{π}{2}$ .

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Step 23.9.1

Multiply $\frac{3}{8}$ by $\frac{π}{2}$ .

$\frac{3 π}{8 \cdot 2}$

Step 23.9.2

Multiply $8$ by $2$ .

$\frac{3 π}{16}$

Step 24

The result can be shown in multiple forms.

Exact Form:

$\frac{3 π}{16}$

Decimal Form:

$0.58904862 \dots$