Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{2}} (1 - \cos (2 t)) \sin (2 t) d t$

Step 1

Let $u_{2} = \cos (2 t)$ . Then $d u_{2} = - 2 \sin (2 t) d t$ , so $- \frac{1}{2} d u_{2} = \sin (2 t) d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \cos (2 t)$ . Find $\frac{d u_{2}}{d t}$ .

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Step 1.1.1

Differentiate $\cos (2 t)$ .

$\frac{d}{d t} [\cos (2 t)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = 2 t$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 t$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d t} [2 t]$

Step 1.1.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d t} [2 t]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $2 t$ .

$- \sin (2 t) \frac{d}{d t} [2 t]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$- \sin (2 t) (2 \frac{d}{d t} [t])$

Step 1.1.3.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 t) \frac{d}{d t} [t]$

Step 1.1.3.3

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

$- 2 \sin (2 t) \cdot 1$

Step 1.1.3.4

Multiply $- 2$ by $1$ .

$- 2 \sin (2 t)$

Step 1.2

Substitute the lower limit in for $t$ in $u_{2} = \cos (2 t)$ .

$u_{lower} = \cos (2 \frac{π}{4})$

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $2$ .

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Step 1.3.1.1

Factor $2$ out of $4$ .

$u_{lower} = \cos (2 \frac{π}{2 (2)})$

Step 1.3.1.2

Cancel the common factor.

$u_{lower} = \cos (2 \frac{π}{2 \cdot 2})$

Step 1.3.1.3

Rewrite the expression.

$u_{lower} = \cos (\frac{π}{2})$

Step 1.3.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $t$ in $u_{2} = \cos (2 t)$ .

$u_{upper} = \cos (2 \frac{π}{2})$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $2$ .

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Step 1.5.1.1

Cancel the common factor.

$u_{upper} = \cos (2 \frac{π}{2})$

Step 1.5.1.2

Rewrite the expression.

$u_{upper} = \cos (π)$

Step 1.5.2

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

$u_{upper} = - \cos (0)$

Step 1.5.3

The exact value of $\cos (0)$ is $1$ .

$u_{upper} = - 1 \cdot 1$

Step 1.5.4

Multiply $- 1$ by $1$ .

$u_{upper} = - 1$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = - 1$

Step 1.7

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

$\int_{0}^{- 1} (1 - u_{2}) \frac{1}{- 2} d u_{2}$

Step 2

Move the negative in front of the fraction.

$\int_{0}^{- 1} (1 - u_{2}) (- \frac{1}{2}) d u_{2}$

Step 3

Multiply $(1 - u_{2}) (- \frac{1}{2})$ .

$\int_{0}^{- 1} 1 (- \frac{1}{2}) - u_{2} (- \frac{1}{2}) d u_{2}$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $1$ .

$\int_{0}^{- 1} - 1 (\frac{1}{2}) - u_{2} (- \frac{1}{2}) d u_{2}$

Step 4.2

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$\int_{0}^{- 1} - (\frac{1}{2}) - u_{2} (- \frac{1}{2}) d u_{2}$

Step 4.3

Multiply $- 1$ by $- 1$ .

$\int_{0}^{- 1} - \frac{1}{2} + 1 u_{2} \frac{1}{2} d u_{2}$

Step 4.4

Multiply $u_{2}$ by $1$ .

$\int_{0}^{- 1} - \frac{1}{2} + u_{2} \frac{1}{2} d u_{2}$

Step 4.5

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

$\int_{0}^{- 1} - \frac{1}{2} + \frac{u_{2}}{2} d u_{2}$

Step 5

Split the single integral into multiple integrals.

$\int_{0}^{- 1} - \frac{1}{2} d u_{2} + \int_{0}^{- 1} \frac{u_{2}}{2} d u_{2}$

Step 6

Apply the constant rule.

$- \frac{1}{2} u_{2}]_{0}^{- 1} + \int_{0}^{- 1} \frac{u_{2}}{2} d u_{2}$

Step 7

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

$- \frac{1}{2} u_{2}]_{0}^{- 1} + \frac{1}{2} \int_{0}^{- 1} u_{2} d u_{2}$

Step 8

By the Power Rule, the integral of $u_{2}$ with respect to $u_{2}$ is $\frac{1}{2} {u_{2}}^{2}$ .

$- \frac{1}{2} u_{2}]_{0}^{- 1} + \frac{1}{2} \frac{1}{2} {u_{2}}^{2}]_{0}^{- 1}$

Step 9

Substitute and simplify.

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Step 9.1

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

$(- \frac{1}{2} \cdot - 1) + \frac{1}{2} \cdot 0 + \frac{1}{2} \frac{1}{2} {u_{2}}^{2}]_{0}^{- 1}$

Step 9.2

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

$- \frac{1}{2} \cdot - 1 + \frac{1}{2} \cdot 0 + \frac{1}{2} (\frac{1}{2} {(- 1)}^{2} - \frac{1}{2} \cdot 0^{2})$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) + \frac{1}{2} \cdot 0 + \frac{1}{2} (\frac{1}{2} {(- 1)}^{2} - \frac{1}{2} \cdot 0^{2})$

Step 9.3.2

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

$\frac{1}{2} + \frac{1}{2} \cdot 0 + \frac{1}{2} (\frac{1}{2} {(- 1)}^{2} - \frac{1}{2} \cdot 0^{2})$

Step 9.3.3

Multiply $\frac{1}{2}$ by $0$ .

$\frac{1}{2} + 0 + \frac{1}{2} (\frac{1}{2} {(- 1)}^{2} - \frac{1}{2} \cdot 0^{2})$

Step 9.3.4

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{2} + \frac{1}{2} (\frac{1}{2} {(- 1)}^{2} - \frac{1}{2} \cdot 0^{2})$

Step 9.3.5

Raise $- 1$ to the power of $2$ .

$\frac{1}{2} + \frac{1}{2} (\frac{1}{2} \cdot 1 - \frac{1}{2} \cdot 0^{2})$

Step 9.3.6

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

$\frac{1}{2} + \frac{1}{2} (\frac{1}{2} - \frac{1}{2} \cdot 0^{2})$

Step 9.3.7

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} + \frac{1}{2} (\frac{1}{2} - \frac{1}{2} \cdot 0)$

Step 9.3.8

Multiply $0$ by $- 1$ .

$\frac{1}{2} + \frac{1}{2} (\frac{1}{2} + 0 (\frac{1}{2}))$

Step 9.3.9

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1}{2} + \frac{1}{2} (\frac{1}{2} + 0)$

Step 9.3.10

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2}$

Step 9.3.11

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

$\frac{1}{2} + \frac{1}{2 \cdot 2}$

Step 9.3.12

Multiply $2$ by $2$ .

$\frac{1}{2} + \frac{1}{4}$

Step 9.3.13

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} \cdot \frac{2}{2} + \frac{1}{4}$

Step 9.3.14

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.14.1

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

$\frac{2}{2 \cdot 2} + \frac{1}{4}$

Step 9.3.14.2

Multiply $2$ by $2$ .

$\frac{2}{4} + \frac{1}{4}$

Step 9.3.15

Combine the numerators over the common denominator.

$\frac{2 + 1}{4}$

Step 9.3.16

Add $2$ and $1$ .

$\frac{3}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$