Reduction formula

TsAmE

Junior Member
Joined
Aug 28, 2010
Messages
55
Use 0π2sinnxdx=n1n0π2sinn2xdx\displaystyle \int_{0}^{\frac{\pi}{2}}sin^{n}xdx = \frac{n - 1}{n} \int_{0}^{\frac{\pi}{2}}sin^{n-2}xdx

to show that, for odd powers of sine,

0π2sin2n+1xdx=2462n357(2n+1)\displaystyle \int_{0}^{\frac{\pi}{2}}sin^{2n + 1}xdx = \frac{2 \cdot 4 \cdot 6 \cdot \cdot \cdot \cdot \cdot 2n }{3\cdot5\cdot7\cdot\cdot\cdot\cdot\cdot(2n + 1)}

Attempt:

I subbed in 2n + 1 into n in the original reduction formula and got:

0π2sin2n+1xdx=2n2n+10π2sin2n1xdx\displaystyle \int_{0}^{\frac{\pi}{2}}sin^{2n+1}xdx = \frac{2n}{2n +1} \int_{0}^{\frac{\pi}{2}}sin^{2n-1}xdx

but I cant see what more I can do
 
I will show for 0π2sinn(x)dx=246(n1)357n\displaystyle \int_{0}^{\frac{\pi}{2}}sin^{n}(x)dx=\frac{2\cdot 4\cdot 6\cdot\cdot\cdot (n-1)}{3\cdot 5\cdot 7\cdot\cdot\cdot n}
and you can try and adapt it to sin2n+1(x)\displaystyle sin^{2n+1}(x). Okey-doke.

0π2sinn(x)dx=1nsinn1(x)cos(x)0π2+n1n0π2sinn2(x)dx=n1n0π2sinn2(x)dx\displaystyle \int_{0}^{\frac{\pi}{2}}sin^{n}(x)dx=\frac{-1}{n}sin^{n-1}(x)cos(x)|_{0}^{\frac{\pi}{2}}+\frac{n-1}{n}\int_{0}^{\frac{\pi}{2}}sin^{n-2}(x)dx=\frac{n-1}{n}\int_{0}^{\frac{\pi}{2}}sin^{n-2}(x)dx

By repeating the application of the above formula:

0π2sinn(x)dx=(n1n)(n3n2)0π2sinn4(x)dx\displaystyle \int_{0}^{\frac{\pi}{2}}sin^{n}(x)dx=\left(\frac{n-1}{n}\right)\left(\frac{n-3}{n-2}\right)\int_{0}^{\frac{\pi}{2}}sin^{n-4}(x)dx

For n odd:

(n1n)(n3n2)(n5n4)230π2sin(x)dxthis equals 1\displaystyle \left(\frac{n-1}{n}\right)\left(\frac{n-3}{n-2}\right)\left(\frac{n-5}{n-4}\right)\cdot\cdot\cdot \frac{2}{3}\underbrace{\int_{0}^{\frac{\pi}{2}}sin(x)dx}_{\text{this equals 1}}

246(n1)357n\displaystyle \frac{2\cdot 4\cdot 6\cdot\cdot\cdot (n-1)}{3\cdot 5\cdot 7\cdot \cdot \cdot n}
 
galactus said:
0π2sinn(x)dx=(n1n)(n3n2)0π2sinn4(x)dx\displaystyle \int_{0}^{\frac{\pi}{2}}sin^{n}(x)dx=\left(\frac{n-1}{n}\right)\left(\frac{n-3}{n-2}\right)\int_{0}^{\frac{\pi}{2}}sin^{n-4}(x)dx

Where did the (n - 3 / n -2) and sin^(n-4)x come from?
 
Top