Calculus Examples

$\int_{\frac{1}{12}}^{\frac{1}{4}} \csc (2 π t) \cot (2 π t) d t$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\csc (2 π t)$ .

$\frac{d}{d t} [\csc (2 π t)]$

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 $f (t) = \csc (t)$ and $g (t) = 2 π t$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$- \csc (u_{1}) \cot (u_{1}) \frac{d}{d t} [2 π t]$

Step 1.1.2.3

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

$- \csc (2 π t) \cot (2 π t) \frac{d}{d t} [2 π t]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$- \csc (2 π t) \cot (2 π t) (2 π \frac{d}{d t} [t])$

Step 1.1.3.2

Multiply $2$ by $- 1$ .

$- 2 \csc (2 π t) \cot (2 π t) (π \frac{d}{d t} [t])$

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 $n = 1$ .

$- 2 \csc (2 π t) \cot (2 π t) (π \cdot 1)$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Multiply $π$ by $1$ .

$- 2 \csc (2 π t) \cot (2 π t) π$

Step 1.1.3.4.2

Reorder the factors of $- 2 \csc (2 π t) \cot (2 π t) π$ .

$- 2 π \cot (2 π t) \csc (2 π t)$

Step 1.2

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

$u_{lower} = \csc (2 π \frac{1}{12})$

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 $2 π$ .

$u_{lower} = \csc (2 (π) \frac{1}{12})$

Step 1.3.1.2

Factor $2$ out of $12$ .

$u_{lower} = \csc (2 (π) \frac{1}{2 (6)})$

Step 1.3.1.3

Cancel the common factor.

$u_{lower} = \csc (2 π \frac{1}{2 \cdot 6})$

Step 1.3.1.4

Rewrite the expression.

$u_{lower} = \csc (π \frac{1}{6})$

Step 1.3.2

Combine $π$ and $\frac{1}{6}$ .

$u_{lower} = \csc (\frac{π}{6})$

Step 1.3.3

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$u_{lower} = 2$

Step 1.4

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

$u_{upper} = \csc (2 π \frac{1}{4})$

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

Factor $2$ out of $2 π$ .

$u_{upper} = \csc (2 (π) \frac{1}{4})$

Step 1.5.1.2

Factor $2$ out of $4$ .

$u_{upper} = \csc (2 (π) \frac{1}{2 (2)})$

Step 1.5.1.3

Cancel the common factor.

$u_{upper} = \csc (2 π \frac{1}{2 \cdot 2})$

Step 1.5.1.4

Rewrite the expression.

$u_{upper} = \csc (π \frac{1}{2})$

Step 1.5.2

Combine $π$ and $\frac{1}{2}$ .

$u_{upper} = \csc (\frac{π}{2})$

Step 1.5.3

The exact value of $\csc (\frac{π}{2})$ is $1$ .

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = 2$

$u_{upper} = 1$

Step 1.7

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

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

Step 2

Move the negative in front of the fraction.

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

Step 3

Apply the constant rule.

$- \frac{1}{2 π} u_{2}]_{2}^{1}$

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $- 1$ by $1$ .

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

Step 4.2.2

Combine $\frac{1}{2 π}$ and $2$ .

$- \frac{1}{2 π} + \frac{2}{2 π}$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

$- \frac{1}{2 π} + \frac{2}{2 π}$

Step 4.2.3.2

Rewrite the expression.

$- \frac{1}{2 π} + \frac{1}{π}$

Step 4.2.4

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

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

Step 4.2.5

Write each expression with a common denominator of $2 π$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.5.1

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

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

Step 4.2.5.2

Reorder the factors of $π \cdot 2$ .

$- \frac{1}{2 π} + \frac{2}{2 π}$

Step 4.2.6

Combine the numerators over the common denominator.

$\frac{- 1 + 2}{2 π}$

Step 4.2.7

Add $- 1$ and $2$ .

$\frac{1}{2 π}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2 π}$

Decimal Form:

$0.15915494 \dots$