Integral of (1 + [1/2x^(1/2) - 1/2x^(-1/2)]^2)^(1/2) from 0 to 3

[carlchen]

from 0 to 3, ( 1 + [ 1/2x^(1/2) - 1/2x^(-1/2) ]^2 )^(1/2) dx

I started to foil the ^2 term and then tryed to use the u sub, but It doesn't seem to work out. this problem is killing me!

 

[HallsofIvy]

Try that again! ((1/2)x^1/2- (1/2)x^-1/2)^1/2=
(1/4)x- 1/2+ (1/4)x^-1.

Now, when you add 1 to that you get (1/4)x+ 1/2+ (1/4)x^-1. In other words, exactly the same thing except with +1/2 instead of -1/2!

Now that = (what?)².

 

[Zurtex]

I think HallsofIvy is saying mess about with the algebra. I did and in a few steps got:

$$\frac{1}{2} \int_0^3 x^{\frac{1}{2}} + x^{-\frac{1}{2}} dx$$

But I am not 100% confident on my skills so don't just copy that down.

 

[e(ho0n3]

Another proof that you need to be good at algebra in order to be good at calculus.

 

[HallsofIvy]

Another proof that you need to be good at algebra in order to be good at calculus.

And you have to be good at calculus to be good at differential equations- and you have to be good at differential equations to be good at analysis, It just keeps going!