Solving Integral: \int{\sqrt{{4-2x^2}}}dx

[pa1o]

Iam having trouble how to solve

$$\int{\sqrt{{4-2x^2}}}dx$$

When I try eulers substitution, the integral gets only terribly complicated and all..

Anyone can help me out with a simple and nice solution ?

 

[dextercioby]

How about
$$\int 2\sqrt{1-(\frac{\sqrt{2}x}{2})^{2}} dx$$

and the substitution
$$\frac{\sqrt{2}x}{2}\rightarrow \sin u$$

Daniel.

 

[HallsofIvy]

I would have done it a slightly different way (mostly because I don't like fractions!): factor out a "2" (instead of "4" as dextecioby does) to get
$$\sqrt{2}\int\sqrt{2- x^2}dx$$
Now let $x= \sqrt{2}sin(\theta)$ so that $2- x^2$ becomes $2- 2sin^2(\theta)$ and $\sqrt{2- x^2}= \sqrt{2(1- sin^2(\theta)}= \sqrt{2 cos^2(\theta)}= \sqrt{2}cos(\theta)$. Of course, $dx= \sqrt{2}cos(\theta)$. You wind up with exactly the same thing as dextercioby's way.

Generally speaking, any time you see something that looks like $\sqrt{1- x^2}$ you should think "$\sqrt{1- sin^2(\theta)}= cos(\theta)$

 

[arildno]

Just a tiny, but rather important correction:
$$\sqrt{1-\sin^{2}\theta}}=|\cos\theta|$$

 

[dextercioby]

How about stating the domain of "x" in the initial problem??In this case all real axis doesn't pose problems,but it could have been only the positive semiaxis and Halls's post would have been flawless (alambicated,but flawless )...




Daniel.

 

[arildno]

WOW!
I've just learned a new English word: "alambicated"
Now, I need to figure out where I can use it..

 

[dextercioby]

I vritually translated the word from Romanian ("alambicat") into English,without really knowing it would exist or not.I searched it with google and came up with 2 references...If the endings ".no (Norway?? )" and ".es" (Spain) are correct,then it's a brand new word in the English language...

Daniel.

P.S.I wish i had invented it...

 

[arildno]

I found the following definition in an on-line dictionary:
"alembic
n. ancient distilling apparatus; purifying or transforming apparatus or act. alembicate, v.t. distil. alembicated, a. rather too refined (of literary style).


© From the Hutchinson Encyclopaedia.
Helicon Publishing LTD 2000.
All rights reserved."

(Still a new word to me, though )

 

[fourier jr]

How about stating the domain of "x" in the initial problem??

wtf why not just leave the | | s ?? it's just a little 1st-year calculus problem. why make things more complicated by doing that

 

[pa1o]

thanks people

THANK YOU all and thank you daniel and all the nice people who helped me )