arctan/arcsin and integration by substitution

$\displaystyle \int \frac{1}{\sqrt{81-x^2}}~dx$

I know I need to use substitution and I am aware of the fact that $\displaystyle \frac{1}{\sqrt{1-x^2}}$ = derivative of arcsin(x), but I am unsure how I apply these to the above problem.

I have the answer $\displaystyle arcsin \left(\frac{x}{9}\right)+C$ but I don't follow the steps to get to it.

Re: arctan/arcsin and integration by substitution

Let $\displaystyle I=\int\frac{1}{\sqrt{81-x^2}}dx$

We can do it by substitution, but I wouldn't personally. However, it's good to start from there.

So I'm going to let $\displaystyle x=9sin(u)$

This means that $\displaystyle \frac{dx}{du}=9cos(u)$ so $\displaystyle dx=9cos(u)~du$

$\displaystyle I=\int\frac{9cos(u)}{\sqrt{81-[9sin(u)]^2}}du$

$\displaystyle =\int\frac{9cos(u)}{\sqrt{81-81sin^2(u)}}du$

Factoring the $\displaystyle 81$:

$\displaystyle =\int\frac{9cos(u)}{\sqrt{81[1-sin^2(u)]}}du$

Applying $\displaystyle \sqrt{a\cdot{b}}=\sqrt{a}\sqrt{b}$ for $\displaystyle a,b\geq{0}$:

$\displaystyle I=\int\frac{9cos(u)}{9\sqrt{1-sin^2(u)}}du$

But as $\displaystyle sin^2(u)+cos^2(u)=1$,

$\displaystyle 1-sin^2(u)=cos^2(u)$

So $\displaystyle I=\int\frac{cos(u)}{\sqrt{cos^2(u)}}du$

$\displaystyle =\int\frac{cos(u)}{cos(u)}du$

$\displaystyle =\int~du$

$\displaystyle =u+C$

Now, I let $\displaystyle x=9sin(u)$ right at the start of my substitution. This means that $\displaystyle \frac{x}{9}=sin(u)$, so $\displaystyle u=arcsin(\frac{x}{9})$

Giving us the required final answer of $\displaystyle I=arcsin(\frac{x}{9})+C$

Is this clear? Usually I'd ask you to do part of the problem yourself but I don't think there's really a good place to stop with this question.

Re: arctan/arcsin and integration by substitution

hi,

the problem is easy

let y=x/9 => x = 9y and dx = 9dy

$\displaystyle \int \frac{9dy}{\sqrt{81-81y^{2}}}$

$\displaystyle \int \frac{9dy}{9\sqrt{1-y^{2}}}$

$\displaystyle \int \frac{dy}{\sqrt{1-y^{2}}}$

then

the solution is : arcsin(y) + c

i.e. arcsin(x/9) + c