Taking Integrals of quantities inside a square root?

Question

Okay, my class is now doing arc lengths and the hardest part of it is taking the integral of a function in the square root. Any function inside the square root with a polynomial greater than 1 is something that cant be done. The only trick i found so far is trying to factor it.

Like one example is taking the integral of say.....

sqrt ((x^2+4x+4)

What i did was change the function and make it look like this....

sqrt ((x + 2)^2)

^^^^ that one is the same as sqrt ((x^2+4x+4)

So the square root and the square cancels out and I just take the integral of (x + 2) instead.

But now, what if you cant do that anymore? If you cant solve it like that anymore?

Is there a way to taking integrals of a function inside a square root? Like say the integral of the square root of (sinx e^x)

How do you do that?

Answer 1

The integral you have given, sqrt(sin (x) * e^x), cannot be solved easily.

There are three main techniques I know of for finding the integral of a function which is something under a square root:

1) Rewrite it as something to make the square cancel.

You have already commented on this technique. For example, if you want to integrate sqrt(x^2 + 4x + 4) dx, you can rewrite it as sqrt((x + 2)^2) dx, which is the same thing as |x + 2| dx. (Or (x + 2) dx, if your limits are greater than or equal to -2.)

2) Use inverse trigonometric substitution.

(This is related to the first way, too--the main point of trigonometric substitution is to get the square root to go away.)

For example, if you want to integrate sqrt(1 - x^2) dx, you can use the substitution x = sin t. This gives you sqrt(1 - sin^2 t) (cos t) dt, which can be rewritten as sqrt(cos^2 t) (cos t) dt, or |cos t| cos t dt, or cos^2 t dt if the limits on your integral happen to be between 0 and pi.

3) Use a u-substitution.

This is less often useful (you'd have to get lucky) and I can't think of an example offhand, but occasionally a clever u-substitution works for this sort of integral.

Answer 2

This Site Might Help You.

RE:
Taking Integrals of quantities inside a square root?
Okay, my class is now doing arc lengths and the hardest part of it is taking the integral of a function in the square root. Any function inside the square root with a polynomial greater than 1 is something that cant be done. The only trick i found so far is trying to factor it.

Like one example...

Answer 3

It's not always doable. It's safe to say that if you're given it as a homework or test problem, there will be a nice answer. But in real life, there are a lot more integrals that don't have elementary solutions.

The methods of solving these are factoring, substitution, trigonometric substitution, integral tables, etc. Again, you just need to recognize that not everything will give you a simple answer.

Btw, in the most general case, you would solve it by using power series, but it won't be an simple-looking answer.

Answer 4

First, one of your statements is not quite true, as
you will learn later on.
If you have to integrate the square root of a
quadratic, you will learn how to use trig
functions to do these. These are expressible
in terms of elementary functions.
But most of the time the integrals you get are
not elementary. That is true.
Even the integral for the arclength of the sine
function turns out to be a type of nonelementary
integral called an elliptic integral.
I ran your example through the wolfram integrator
and it couldn't find a formula for it.

Answer 5

V(x^2 + 4x + 4) = |x+2|

Not all the functions can be integrated, only a few ones can. In case of needing to integrate a non integrable function, you can always use the Taylor series.

Ilusion

Answer 6

Hehehe. If taking integrals were easy, any fool could do it ☺
Seriously, one of the most powerful tools in your arsenal (aside from a good table of 'standard form' integrals) is the metod of integration by parts.
⌠u∙dv = uv - ⌠v∙du
The only thing is in deciding what to use for u and dv. And the only answer is lots of practice, a bit of insight, and a heaping cupful of good luck.
BTW, the integral you asked about [√(sin(x)e^x)] is a standard form that leads to a fairly complicated little thing called a 'hypergeometric function' and it's one that you'll get to soon enough. Trust me ☺

Doug

Answer 7

integrals quantities square root

Answer 8

There's no universal rule for dealing with problems like this. In this particular case, you could try splitting this up into (√sin(x)) e^(x/2), then try integration by parts.

Also, keep in mind the derivatves for the inverse trig functions. Some of them involve square roots.

Answer 9

http://www.mathalino.com/reviewer/engineering-mechanics/706-centroiud-quarter-circle-integration

How is the root expression in the 3rd step changed to the 3/2 expression. Can anyone please give a detail.