# Thread: "u sub" problem with no "u"

1. ## "u sub" problem with no "u"

I need to know how to integrate 2sqrt(100-x^2) from -10 to 10 by hand. It suggests a u-substitution but no u seems to exist. Please help.

Bret Norvilitis
Orchard Park HS

2. Originally Posted by bmnorvil
I need to know how to integrate 2sqrt(100-x^2) from -10 to 10 by hand. It suggests a u-substitution but no u seems to exist. Please help.

Bret Norvilitis
Orchard Park HS
It has been a while since I have done any integration by substitution but I would maybe suggest using $u = 100-x^2$

3. I was never really that sure about substitution but i think this is how it works...

When it says to do a u-sub it means set x to be equal to some function containing u. This also means that you should change your limits as well.

In this case, set $x = 10\sin(u)$ and hence $dx = 10\cos(u)du$ then $x^2 = 100\sin^2(u)$.

So put this into your equation and you get...

$\int 2 \sqrt(100-x^2)dx = \int 2 \sqrt(100-100\sin^2(u))du = \int 2 \sqrt(100(1-\sin^2(u))) 10\cos(u)du$.
= $\int 2 \sqrt(100cos^2(u)) 10\cos(u)$.
= $\int 200cos^2(u)$

4. let 10sinu=x
10cosu(du)=dx
du=dx/(10cosu)

cosu=(100-x^2)^1/2/10

5. ^^^^^^
What Amer said.

Note that Amers integral limits were found by setting 10 and -10 to be equal to $10\sin(u)$, hence \sin(u) = 1 and -1, so $u= \pm \frac{\pi}{2}$ are the new limits