Notes: `int cot^4(theta) d theta` Use integration tables to find the indefinite integral.

Tuesday, 30 May 2017

$\displaystyle\int{{\cot}}^{{4}}{\left(\theta\right)}{d}\theta$ Use integration tables to find the indefinite integral.

Indefinite integral follows the formula: $\displaystyle\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({x}\right)}+{C}$

where:

$\displaystyle{f{{\left({x}\right)}}}$ as the integrand function

$\displaystyle{F}{\left({x}\right)}$ as the antiderivative of $\displaystyle{f{{\left({x}\right)}}}$

$\displaystyle{C}$ as constant of integration.

The given integral problem: $\displaystyle\int{{\cot}}^{{4}}{\left(\theta\right)}{d}\theta$ resembles one of the formulas from the integration table. It follows the integration formula for cotangent function as :

$\displaystyle\int{{\cot}}^{{n}}{\left({x}\right)}{\left.{d}{x}\right.}=-\frac{{{{\cot}}^{{{\left({n}-{1}\right)}}}{\left({x}\right)}}}{{{n}-{1}}}-\int{{\cot}}^{{{\left({n}-{2}\right)}}}{\left({x}\right)}{\left.{d}{x}\right.}$ .

Applying the formula, we get:

$\displaystyle\int{{\cot}}^{{4}}{\left(\theta\right)}{d}\theta\ldots$

Indefinite integral follows the formula: $\displaystyle\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({x}\right)}+{C}$

where:

$\displaystyle{f{{\left({x}\right)}}}$ as the integrand function

$\displaystyle{F}{\left({x}\right)}$ as the antiderivative of $\displaystyle{f{{\left({x}\right)}}}$

$\displaystyle{C}$ as constant of integration.

Applying the formula, we get:

$\displaystyle\int{{\cot}}^{{4}}{\left(\theta\right)}{d}\theta=-\frac{{{{\cot}}^{{{\left({4}-{1}\right)}}}{\left(\theta\right)}}}{{{4}-{1}}}-\int{{\cot}}^{{{\left({4}-{2}\right)}}}{\left(\theta\right)}{d}\theta$

$\displaystyle=-\frac{{{{\cot}}^{{3}}{\left(\theta\right)}}}{{3}}-\int{{\cot}}^{{2}}{\left(\theta\right)}{d}\theta$

To further evaluate the integral part: $\displaystyle\int{{\cot}}^{{2}}{\left(\theta\right)}{d}\theta$ we may apply trigonometric identity: $\displaystyle{{\cot}}^{{2}}{\left(\theta\right)}={{\csc}}^{{2}}{\left(\theta\right)}-{1}$ .

$\displaystyle\int{{\cot}}^{{2}}{\left(\theta\right)}{d}\theta=\int{\left[{{\csc}}^{{2}}{\left(\theta\right)}-{1}\right]}{d}\theta$

Apply basic integration property: $\displaystyle\int{\left({u}-{v}\right)}{\left.{d}{x}\right.}=\int{\left({u}\right)}{\left.{d}{x}\right.}-\int{\left({v}\right)}{\left.{d}{x}\right.}.$

$\displaystyle\int{\left[{{\csc}}^{{2}}{\left(\theta\right)}-{1}\right]}{d}\theta=\int{{\csc}}^{{2}}{\left(\theta\right)}{d}\theta-\int{1}{d}\theta$

$\displaystyle=-{\cot{{\left(\theta\right)}}}-\theta+{C}$

Note: From basic integration property: $\displaystyle\int{\left.{d}{x}\right.}={x}$ then $\displaystyle\int{1}{d}\theta=\int{d}\theta=\theta$ .

From the integration table for trigonometric function, we have $\displaystyle\int{{\csc}}^{{2}}{\left({x}\right)}{\left.{d}{x}\right.}=-{\cot{{\left({x}\right)}}}$ then $\displaystyle\int{{\csc}}^{{2}}{\left(\theta\right)}{d}\theta=-{\cot{{\left(\theta\right.}}}$ ).

applying $\displaystyle\int{\left[{{\cot}}^{{2}}{\left(\theta\right)}\right]}{d}\theta=-{\cot{{\left(\theta\right)}}}-\theta+{C}$ , we get the complete indefinite integral as:

$\displaystyle\int{{\cot}}^{{4}}{\left(\theta\right)}{d}\theta=-\frac{{{{\cot}}^{{3}}{\left(\theta\right)}}}{{3}}-\int{{\cot}}^{{2}}{\left(\theta\right)}{d}\theta$

$\displaystyle=-\frac{{{{\cot}}^{{3}}{\left(\theta\right)}}}{{3}}-{\left(-{\cot{{\left(\theta\right)}}}-\theta\right)}+{C}$

$\displaystyle=-\frac{{{{\cot}}^{{3}}{\left(\theta\right)}}}{{3}}+{\cot{{\left(\theta\right)}}}+\theta+{C}$

Notes

Tuesday, 30 May 2017

$\displaystyle\int{{\cot}}^{{4}}{\left(\theta\right)}{d}\theta$ Use integration tables to find the indefinite integral.

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How are race, gender, and class addressed in Oliver Optic's Rich and Humble?

Tuesday, 30 May 2017

Use integration tables to find the indefinite integral.

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Post a Comment

How are race, gender, and class addressed in Oliver Optic&#39;s Rich and Humble?

$\displaystyle\int{{\cot}}^{{4}}{\left(\theta\right)}{d}\theta$ Use integration tables to find the indefinite integral.

How are race, gender, and class addressed in Oliver Optic's Rich and Humble?