Integration by substitution (I think)

1. Aug 30, 2011

tomwilliam

1. The problem statement, all variables and given/known data

Integral of d.cos j with regard to d.sin j

Where d is a constant.

2. Relevant equations

3. The attempt at a solution
I don't know how to approach this. I can substitute u=d.sin j
Then I have
Integral of dz/dj with regard to dz, but not sure where to go from here.
Any help appreciated.

2. Aug 30, 2011

lanedance

do you mean a Riemann–Stieltjes integral?
$$\int f(j) dg(j)$$

with
$$\int f(j) = sin(u)$$
$$\int f(j) dg(j) = cos(u)$$

First its probably a bad idea to use d as a symbol for constant in this context, based on its calculus context

Now if f and g has a continuous bounded derivative in a Riemann–Stieltjes integral the following equality holds
$$\int_a^b f(j) dg(j) = \int_a^b f(j) g'(j) dj$$

Last edited: Aug 30, 2011
3. Aug 30, 2011

tomwilliam

Thanks,
Ok, the complexity of your answer tells me I've made an earlier mistake. I was trying to integrate the expression

sqrt(a^2 - b^2)

With regard to b. I used the substitution b=a sin theta so that

Sqrt(a^2(1-sin^2 theta) = sqrt(a^2 cos^2 theta) = a cos theta

Now I have to integrate

a cos theta

With regard to a sin theta. I've changed the variables, but I think that's equivalent to my original post. Did I make a mistake?

4. Aug 30, 2011

lanedance

ahh ok so you mean
$$\int \sqrt{a^2-b^2}db$$

now let b = a.sin(t)
$$b = a.sin(t)$$
$$db = a.cos(t).dt$$

subbing in
$$\int \sqrt{a^2-a^2 sin^2(t)}a.cos(t).dt$$
$$\int \sqrt{a^2(1- sin^2(t))}a.cos(t).dt$$
$$\int \sqrt{a^2cos^2(t)}a.cos(t).dt$$

so the integral should be with respect to t (short for theta)

5. Aug 30, 2011

lanedance

i always do those steps with b & db to keep it clear, similar with limits if the integral has limits

6. Aug 30, 2011

lanedance

by the way if you ever want to write tex, you can right click on the expression to show source and see how its written

7. Aug 30, 2011

tomwilliam

Thanks, that's exactly it.
I'm writing on an iPad, which makes it difficult (impossible?) to type tex without putting it into a separate application first.

8. Aug 30, 2011

lanedance

yeah know the feeling

9. Aug 30, 2011

tomwilliam

EDIT: It's ok, i've solved it now!

Last edited: Aug 30, 2011