Calculus Examples

$\int_{\frac{1}{6}}^{\frac{1}{2}} \csc (π A) \cot (π A) d A$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\csc (π A)$ .

$\frac{d}{d A} [\csc (π A)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d A} [f (g (A))]$ is $f' (g (A)) g' (A)$ where $f (A) = \csc (A)$ and $g (A) = π A$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

$- \csc (π A) \cot (π A) (π \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $π$ by $1$ .

$- \csc (π A) \cot (π A) π$

Step 1.1.3.3.2

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

$- π \cot (π A) \csc (π A)$

Step 1.2

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

$u_{lower} = \csc (π \frac{1}{6})$

Step 1.3

Simplify.

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Step 1.3.1

Combine $π$ and $\frac{1}{6}$ .

$u_{lower} = \csc (\frac{π}{6})$

Step 1.3.2

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

$u_{lower} = 2$

Step 1.4

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

$u_{upper} = \csc (π \frac{1}{2})$

Step 1.5

Simplify.

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Step 1.5.1

Combine $π$ and $\frac{1}{2}$ .

$u_{upper} = \csc (\frac{π}{2})$

Step 1.5.2

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.

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

Step 2

Dividing two negative values results in a positive value.

$\int_{2}^{1} - \frac{1}{π} d u_{2}$

Step 3

Apply the constant rule.

$- \frac{1}{π} u_{2}]_{2}^{1}$

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Multiply $- 1$ by $1$ .

$- \frac{1}{π} + \frac{1}{π} \cdot 2$

Step 4.3

Combine $\frac{1}{π}$ and $2$ .

$- \frac{1}{π} + \frac{2}{π}$

Step 4.4

Simplify.

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Step 4.4.1

Combine the numerators over the common denominator.

$\frac{- 1 + 2}{π}$

Step 4.4.2

Add $- 1$ and $2$ .

$\frac{1}{π}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{π}$

Decimal Form:

$0.31830988 \dots$