Evaluate the following indefinite integral. \int \cos^{4}(x)\sin(x)dx

Judith McQueen 2022-01-07 Answered
Evaluate the following indefinite integral.
\(\displaystyle\int{{\cos}^{{{4}}}{\left({x}\right)}}{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

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Expert Answer

macalpinee3
Answered 2022-01-08 Author has 4107 answers

Step 1
See attached file for a step by step explanation to find the given integral.
Step 2
Solution \(\displaystyle\int{{\cos}^{{{4}}}{\left({x}\right)}}{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}\)
\(\displaystyle\Rightarrow\int{\left({\cos{{\left({x}\right)}}}\right)}^{{{4}}}{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}\)
Let \(\cos(x)=t\)
differentiate \(\displaystyle-{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}={\left.{d}{t}\right.}\)
\(\displaystyle{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}=-{\left.{d}{t}\right.}\)
\(\displaystyle\Rightarrow\int{t}^{{{4}}}{\left(-{\left.{d}{t}\right.}\right)}\)
\(\displaystyle\Rightarrow-\int{t}^{{{4}}}{\left.{d}{t}\right.}\)
\(\displaystyle\Rightarrow-{\frac{{{t}^{{{5}}}}}{{{5}}}}+{c}\)
put \(\displaystyle{t}={\cos{{\left({x}\right)}}}\)
\(\displaystyle\Rightarrow-{\frac{{{{\cos}^{{{5}}}{x}}}}{{{5}}}}+{c}\) or \(\displaystyle-{\frac{{{\left({\cos{{x}}}\right)}^{{{5}}}}}{{{5}}}}+{c}\)

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Jenny Sheppard
Answered 2022-01-09 Author has 2393 answers
\(\displaystyle{\left(-{\sin{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}={d}{\left({\cos{{\left({x}\right)}}}\right)},{t}={\cos{{\left({x}\right)}}}\)
\(\displaystyle\int{\left(-{t}^{{{4}}}\right)}{\left.{d}{t}\right.}\)
This is a tabular integral:
\(\displaystyle\int-{t}^{{{4}}}{\left.{d}{t}\right.}=-{\frac{{{t}^{{{5}}}}}{{{5}}}}+{C}\)
To write down the final answer, it remains to substitute \(\displaystyle{\cos{{\left({x}\right)}}}\) instead of t.
Result:
\(\displaystyle-{\frac{{{\cos{{\left({x}\right)}}}^{{{5}}}}}{{{5}}}}+{C}\)
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karton
Answered 2022-01-11 Author has 8659 answers

\(\\\text{We have:} \\\int \cos(x)^{4}\sin(x)dx \\\int -t^{4}dt \\-\int t^{4}dt \\\text{Evaluate} \\-\frac{t^{5}}{5} \\-\frac{\cos(x)^{5}}{5} \\\text{Answer:} \\-\frac{\cos(x)^{5}}{5}+C\)

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