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[logistic_guy]

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

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

 

[khansaheb]

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

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

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[logistic_guy]

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

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

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

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

Then,

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


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


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

Then,

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