Evaluate the following intcsc^6udu

Evaluate the following intcsc^6udu

Question
Trigonometry
asked 2021-03-09
Evaluate the following
\(\displaystyle\int{{\csc}^{{6}}{u}}{d}{u}\)

Answers (1)

2021-03-10
The trigonometric function \(\displaystyle\int{{\csc}^{{6}}{u}}{d}{u}\)
\(\displaystyle\int{{\csc}^{{6}}{u}}{d}{u}=\int{{\csc}^{{4}}{u}}\cdot{{\csc}^{{2}}{u}}{d}{u}\)
\(\displaystyle=\int{\left({{\csc}^{{2}}{u}}\right)}^{{2}}\cdot{{\csc}^{{2}}{u}}{d}{u}\)
\(\displaystyle=\int{\left({1}+{{\cot}^{{2}}{u}}\right)}^{{2}}\cdot{{\csc}^{{2}}{u}}{d}{u}\)
(Since, \(\displaystyle{{\csc}^{{2}}{u}}={1}+{{\cot}^{{2}}{u}}\))
Let \(\displaystyle{t}={\cot{{u}}}\)
\(\displaystyle{\frac{{{\left.{d}{t}\right.}}}{{{d}{u}}}}=-{{\csc}^{{2}}{u}}\) implies \(\displaystyle-{\left.{d}{t}\right.}={{\csc}^{{2}}{u}}{d}{u}\)
Now, \(\displaystyle\int{{\csc}^{{6}}{d}}{u}=-\int{\left({1}+{t}^{{2}}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle=-\int{\left({1}+{2}{t}^{{2}}+{t}^{{4}}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle=-{\left({t}+{\frac{{{2}{t}^{{3}}}}{{{3}}}}+{\frac{{{t}^{{5}}}}{{{5}}}}\right)}+{C}\)
\(\displaystyle=-{t}-{\frac{{{2}{t}^{{3}}}}{{{3}}}}-{\frac{{{t}^{{5}}}}{{{5}}}}+{C}\)
Here \(\displaystyle{t}={\cot{{u}}}\)
Therefore, \(\displaystyle\int{{\csc}^{{6}}{u}}{d}{u}=-{\cot{{u}}}-{\frac{{{2}}}{{{3}}}}{{\cot}^{{3}}{u}}-{\frac{{{1}}}{{{5}}}}{{\cot}^{{5}}{u}}+{C}\)
0

Relevant Questions

asked 2021-02-24
Evaluate the following.
\(\displaystyle\int{{\sin}^{{2}}{x}}\cdot{\tan{{x}}}{\left.{d}{x}\right.}\)
asked 2020-12-15
Evaluate the following.
\(\displaystyle\int{\cos{\beta}}{\left({1}-{\cos{{2}}}\beta\right)}^{{3}}{d}\beta\)
asked 2021-02-25
Evaluate the following.
\(\displaystyle\int{\frac{{{\left({10}{y}+{11}\right)}{\left.{d}{y}\right.}}}{{{4}{y}^{{2}}-{4}{y}+{5}}}}\)
asked 2020-10-25
Evaluate the following.
\(\displaystyle\int{\frac{{{{\cos}^{{5}}{\left({3}{z}\right)}}{\left.{d}{z}\right.}}}{{{{\sin}^{{2}}{\left({3}{z}\right)}}}}}\)
asked 2021-02-09
Evaluate the following.
\(\displaystyle\int{\frac{{{\sin{\theta}}{\left({\cos{\theta}}+{4}\right)}{d}\theta}}{{{1}+{{\cos}^{{2}}\theta}}}}\)
asked 2021-02-08
Explain how we can evaluate the expression \(\displaystyle{\cos{{\left({\arctan{{\left({\frac{{{v}}}{{{a}}}}\right)}}}\right)}}}\) and what the evaluation is in terms of v and a.
asked 2020-12-25
Find the value of each trigonometric ratio to the nearest ten-thousandth
\(\displaystyle{{\tan{{50}}}^{\circ}}\)
asked 2020-10-18
Determine the exact value of expression.
\(\displaystyle{\sec{{\left({210}\right)}}}\times{\cot{{\left({300}\right)}}}+{\sin{{\left({225}\right)}}}\)
asked 2020-12-25
Determine the exact value of expression.
\(\displaystyle{\tan{{\left({60}\right)}}}\times{3}{\sin{{\left({90}\right)}}}-{\sin{{\left({315}\right)}}}\)
asked 2021-02-05
Determine the exact value of expression.
\(\displaystyle{\tan{{\left({330}\right)}}}\times{\cos{{\left({240}\right)}}}-{2}{\cos{{\left({270}\right)}}}\)
...