# Integral help from S.R. and G.R. physicsforums

1. Mar 1, 2008

### morrobay

Would someone evaluate this integral:
See Physics Forums -Special and General Relativity above.
On page one with title: Elapsed time on accelerating clock. by morrobay.
In particular reply # 4 by JesseM .
I seem to be missing something with The Integrator reference.
And one semester calculus , (Thomas ) is not enough.
thanks

2. Mar 2, 2008

### Gib Z

Try a simple substitution, $$x=\frac{\cos u}{\sqrt{a}}$$

3. Mar 3, 2008

### morrobay

Integral help from S.R. and G.R physicsforums

If that substitution is simple for you or anyone else I would like to see it

4. Mar 3, 2008

### HallsofIvy

Staff Emeritus
Then look in any introductory calculus book!

Since cos2(u)= 1- sin2(u), setting x= sin(u) is a "natural" choice for any integrand of the form $\sqrt{1- x^2}dx$. With x= sin(u), that $\sqrt{1- x^2}= \sqrt{1- sin^2(u)}= cos(u)$ and dx= cos(u)du. $\int \sqrt{1- x^2} dx= \int cos^2(u)du$ which can be integrated using the trig identity "cos2(u)= (1/2)(1+ cos(2u))".

If you have integrand $\sqrt{a^2- x^2}$ just factor out the "a2": $a\sqrt{1- x^2/a^2}$ and obvious substitution is x/a= sin(u).

5. Mar 3, 2008

### Gib Z

$$\int \sqrt{ 1- ax^2} dx$$

Make the substitution from the previous post.

Then $$dx = - \frac{\sin u}{\sqrt{a}}$$.

The integral is then transformed into;
$$\int \sqrt{ 1- a \cdot \frac{\cos^2 u}{a} } ( - \frac{\sin u}{\sqrt{a}}) du$$.

Constants may be taken out of an integral, and $$1 - \cos^2 u = \sin^2 u$$ by the Pythagorean Identity.

$$-\frac{1}{\sqrt{a}} \int \sin^2 u du$$

$$\sin^2 u = \frac{1 - \cos (2u)}{2}$$ which is verifiable by the double angle identity; $\cos (2t) = \cos^2 t - \sin^2 t$.

$$-\frac{1}{\sqrt{a}} \int \left( \frac{1}{2} - \frac{2 \cos (2u)}{4} du \right)$$

Split the integral and in the second one, let w= 2u, so dw= 2 du.
$$= -\frac{1}{\sqrt{a}} \left( \frac{u}{2} - \frac{\sin (2u)}{4} \right)$$

where $$x= \frac{\cos u}{\sqrt{a}}$$, or $$u = \arccos \left( \sqrt{a}u\right)$$

EDIT: Just saw halls post, should have chosen his substitution because the signs work out more nicely, but this still works out fine.

Last edited: Mar 3, 2008
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