The Notes: `int cot^5(theta) sin^4(theta) d theta` Evaluate the integral

$\displaystyle\int{{\cot}}^{{5}}{\left(\theta\right)}{{\sin}}^{{4}}{\left(\theta\right)}{d}\theta=$

$\displaystyle\int{\left[\frac{{{{\cos}}^{{5}}{\left(\theta\right)}}}{{{{\sin}}^{{5}}{\left(\theta\right)}}}\right]}{{\sin{{t}}}}^{{4}}{\left(\theta\right)}{d}\theta=$

$\displaystyle\int{\left[\frac{{{{\cos}}^{{5}}{\left(\theta\right)}}}{{{{\sin}}^{{\theta}}}}\right]}{d}{\left(\theta\right)}=$

$\displaystyle\int{{\cos}}^{{4}}{\left(\theta\right)}{\left[\frac{{\cos{{\left(\theta\right)}}}}{{\sin{{\left(\theta\right)}}}}\right]}{d}{\left(\theta\right)}=$

$\displaystyle\int{{\left({{\cos}}^{{2}}{\left(\theta\right)}\right)}}^{{2}}{\left[\frac{{\cos{{\left(\theta\right)}}}}{{\sin{{\left(\theta\right)}}}}\right]}{d}{\left(\theta\right)}=$

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

Integrate using u-subsitution.

Let

$\displaystyle{u}={\sin{{\left(\theta\right)}}}$

$\displaystyle\frac{{{d}{u}}}{{{d}{\left(\theta\right)}}}={\cos{{\left(\theta\right)}}}$

$\displaystyle{d}{\left(\theta\right)}=\frac{{{d}{u}}}{{{\cos{{\left(\theta\right)}}}}}$

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

$\displaystyle\int\frac{{{1}-{2}{{u}}^{{2}}+{{u}}^{{4}}}}{{u}}{d}{u}=$

$\displaystyle\int{\left[{\left(\frac{{1}}{{u}}\right)}-{2}{u}+{{u}}^{{3}}\right]}{d}{u}=$

$\displaystyle{\ln}{\left|{u}\right|}-\frac{{{2}{{u}}^{{2}}}}{{2}}+\frac{{{u}}^{{4}}}{{4}}+{C}=$

$\displaystyle{\ln}{\left|{u}\right|}-{{u}}^{{2}}+\frac{{1}}{{4}}{{u}}^{{4}}+{C}=$

$\displaystyle{\ln}{\left|{\sin{{\left(\theta\right)}}}\right|}-{{\sin}}^{{2}}{\left(\theta\right)}+\frac{{1}}{{4}}{{\sin}}^{{4}}{\left(\theta\right)}+{C}$

The final answer is: $\displaystyle{\ln}{\left|{\sin{{\left(\theta\right)}}}\right|}-{{\sin}}^{{2}}{\left(\theta\right)}+\frac{{1}}{{4}}{{\sin}}^{{4}}{\left(\theta\right)}+{C}$

The Notes

Monday, September 7, 2015

$\displaystyle\int{{\cot}}^{{5}}{\left(\theta\right)}{{\sin}}^{{4}}{\left(\theta\right)}{d}\theta$ Evaluate the integral

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