Re: [MATHEDCC] Integration question

mark snyder (msnyder@TIAC.NET)
Tue, 3 Mar 1998 17:37:12 -0500

At 2:05 PM -0500 3/3/98, Bret Taylor wrote:
>Mathematical, not cultural.
>
>Has anyone ever seen or used integration by parts to derive the formula for
>THE INTEGRAL OF SINE SQUARED X ? Hope that makes sense. I'll try it this
>way:
>
>Integral of (sin x)^2 dx
>
>I was just playing around trying to come up with test questions on
>techniques of integration and noticed that the square of the sine of x could
>be integrated using this technique. It's rather nice, complicateed but not
>overly so - needing to use integration by parts twice and also remembering
>some trig formulas for double angles and some basic algebra.

(snip)

Neat. You can also get it by integrating by parts just once:

Int(sinx.sinx dx) = sinx.(-cosx) + Int(cos^2 dx)

(using u = sinx, dv = sinx dx).

Then use cos^2 = 1 - sin^2, so you get:

2*Int(sin^2 dx) = Int(1.dx) - sinx.cosx

Int(sin^2 dx) = (1/2)*x - (1/4)*sin2x

(using sin2x = 2 sinx cosx).

mark snyder

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