How Do You Integrate the Function \(\frac{1}{\sqrt{a^2 - x^2}}\)?

[lazypast]

hi, I've a function here and I am finding it hard to integrate

$$\int \frac {1}{\sqrt{a^2 - x^2}}$$

let x = asin(theta) so $$\frac{dx}{d \theta}=acos \theta$$


$$dx=acos \theta d \theta$$

$$\int \frac {acos \theta d \theta}{\sqrt{a^2 - a^2 sin^2 \theta}}$$

$$\int \frac {acos \theta d \theta}{\sqrt{a^{2} (1 - sin^2 \theta)}}$$

$$\int \frac {acos \theta d \theta}{acos \theta} = \int d \theta$$

i can't continue this as i can't get the right equation which is able to be integrated. any ideas?

 

[d_leet]

What is the problem you're having, you reduced a diffcult integral into a practically trivial one?

 

[lazypast]

i reduced it to dtheta, but I am unable to sub anything back into the integral which i can solve.

$$d \theta = \frac {dx}{acos \theta}$$


$$\int \frac {dx}{acos \theta}$$

but this isn't any more help to me, i know what the answr is ( sin^-1 (x/a)) but arent close to it

 

[d_leet]

Don't substitute back yet, that defeats the whole point of the substitution. Just integrate d(theta).

 

[TMM]

And THEN substitute back in with a reference triangle or just theta in terms of x.

That's how trig substitution works.

 

[Gib Z]

Would it help if you rewrote it as $$\int 1 d\theta$$? What (family of) function(s) whose derivative with respect to theta gives 1? As d_leet pointed out, you really already did the hard part, Made some nice soup but don't know how to use a spoon :(

 

[HallsofIvy]

This also has nothing to do with Differential Equations so I'm moving it to "Calculus and Analysis".

 

[lazypast]

$$\int d \theta = \theta$$

$$\frac {dx}{d \theta} = acos \theta$$

$$\theta = cos^{-1}(\frac {1}{a} \frac {dx}{d \theta})$$

and this still has a dx/dtheta in it. i really am stumpted here

 

[lazypast]

aha I've been abit blind.

$$x = asin \theta$$

$$\theta = sin^{-1} \frac {x}{a}$$

Would it help if you rewrote it as $$\int 1 d\theta$$? What (family of) function(s) whose derivative with respect to theta gives 1? As d_leet pointed out, you really already did the hard part, Made some nice soup but don't know how to use a spoon :(

i don't understand the way you show it.
when f(x) = theta, f'(x) = 1 when you differentiate with respect to theta

thanks for the help anyway

 

[Gib Z]

i don't understand the way you show it.
when f(x) = theta, f'(x) = 1 when you differentiate with respect to theta

Are you familiar with the fundamental theorem on calculus?

 

[lazypast]

i haven't been tought the fundamental theory of calculus,but i can do integration and differentiation which is what i thought is basically was

 

[Gib Z]

:( There are a few versions of the theorem, and they are all equivalent, you really need to know the theorem, and I'm sure you already do but just don't realize it.

It basically says that $$\int^b_a f(x) dx = F(b) - F(a) , \mbox{where} F'(x)=f(x)$$, or equivalently, $$\frac{d}{dx}\int^x_a f(t) dt = f(x)$$. Try and see why they are the same.

 

[ZioX]

What's the derivative of theta with respect to theta?

 

[lazypast]

derivative of theta with respect to theta id say is one.

after looking at some concepts of the fundamental theorem of calculus i can't seem to understand them. you say

$$\int^b_a f(x) dx = F(b) - F(a) , \mbox{where} Fsingle-quote(x)=f(x)$$

i would have thought F(x) = f ' (x)

i don't know why you use F and f

if you replace f(x) with y, then i suppose F(x) could be z?

 

[Gib Z]

It doesn't matter what letters you use, the theorem stays the same though :( It is F'(x) = f(x), you really need to learn this, check your textbook.

 

[HallsofIvy]

$$\int d \theta = \theta$$

$$\frac {dx}{d \theta} = acos \theta$$

$$\theta = cos^{-1}(\frac {1}{a} \frac {dx}{d \theta})$$

and this still has a dx/dtheta in it. i really am stumpted here

aha I've been abit blind.

$$x = asin \theta$$

$$\theta = sin^{-1} \frac {x}{a}$$


thanks for the help anyway

These were posts 8 and 9 but we went on for 6 more posts anyway!

(Oops, 7 now.)

 

[Gib Z]

Well its because we found a serious problem in his understanding :( He says he knows differential and integral calculus, yet has never heard of the Fundamental Theorem before!

 

[lazypast]

i have heard of it just never been tought it. I've just found out that $$\int f(x) = F(x)$$ and that was the notation i didnt understand. like the capital F had been plucked from no where