Re: Two little integrals


"novelo" <novelo@xxxxxxxxx> wrote in message
news:1121035931.645124.110830@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> Hello everybody. I'm stucked with two integrals.
> I was trying to solve an integral by trigonometrical substitution:
>
> 1
> Integral of: ------------
> (36-x^2)^(4/2)
>
> This is the integral of one over the square root of ((36 - x^2)^4)
>
> Well... so I did all the trigonometric substitution and I get the
> following integral:
>
> 1
> --- Integral of: (secant of theta)^3 dtheta
> 216
>
> My problem is that I cannot find an easy way to do the integral of (sec
> x)^3. I've tryed every trigonometric property I know, integration by
> parts, etc... and I get cyclic results that bring me nowhere.
>
> Can someone tell me an easy method to solve this integral? My
> calculator gets me a very long answer, like 6 additions so, is there
> any way or are there any trigonometric properties to solve in an easier
> way this integral? If not, how does the calculator gives so big
> answers?
>
> An the other one that I've tryed for days and I don't get even near to
> the response (by the way I've tryed this one in 4 calculators and none
> was able to do it) is:
>
> Square root of: [sin(square root of x)] --> This one is not urgent, it
> came outfor curiosity only, but the other one must be solvable, but,
> how!?
>
> Thank you!
>

The integral of (sec(theta)) ^ 3 is obtained by integration by parts, but it
comes up so often that it's worth memorizing. That integral is (1/2)
(sec(theta))(tan(theta) - (1/2)ln | sec(theta) + tan(theta) | + C.

As far as the second one, I don't think you can do it in terms of elementary
terms; at least I don't see a way.

Dave


.

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