삼각함수의 적분

월리스 곱 (Wallis product formula)

 

\(\int\sin^n {x}\;dx = -\frac{\sin^{n-1} x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2} x\;dx \qquad\mbox{(for }n>2\mbox{)}\,\!\)

 

(증명)

 

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

 

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

치환적분 \(u=\cos x\), \(dv=\sin^{n-2}x\cos x \dx\)

 

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

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

 

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

 

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\(\int\cos^n x\;dx = \frac{\cos^{n-1} x\sin x}{n} + \frac{n-1}{n}\int\cos^{n-2} x\;dx \qquad\mbox{(for }n>2\mbox{)}\,\!\)

 

 

http://en.wikipedia.org/wiki/Integral_of_secant_cubed