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[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{Solve.}}\)

\(\displaystyle \int \cos^n x \ dx\)

 

[khansaheb]

\(\displaystyle \textcolor{indigo}{\bold{Solve.}}\)

\(\displaystyle \int \cos^n x \ dx\)

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[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{Solve.}}\)

\(\displaystyle \int \cos^n x \ dx\)

\(\displaystyle \int \cos^n x \ dx = \int \cos^{n-1}x \cos x \ dx\)

\(\displaystyle u = \cos^{n-1}x\)
\(\displaystyle du = -(n - 1)\cos^{n-2}x \sin x \ dx\)
\(\displaystyle dv = \cos x \ dx\)
\(\displaystyle v = \sin x\)

Then,

\(\displaystyle \int \cos^n x \ dx = \cos^{n-1}x \sin x + (n - 1)\int \cos^{n-2}x \sin^2 x \ dx\)


\(\displaystyle = \cos^{n-1}x \sin x + (n - 1)\int \cos^{n-2}x (1 - \cos^2 x) \ dx\)


\(\displaystyle = \cos^{n-1}x \sin x + (n - 1)\int (\cos^{n-2}x - \cos^n x) \ dx\)

Then,

\(\displaystyle \int \cos^n x \ dx = \frac{1}{n}\cos^{n-1}x \sin x + \frac{n - 1}{n}\int \cos^{n-2}x \ dx\)