# Solve this integral

1. Jun 16, 2004

### jcsd

Can someone solve this integral as the answer I get looks suspicously complicated:

$$\int^{0}_{\frac{-u}{a}} t\sqrt{1 - \frac{(u + at)^2}{c^2}} dt$$

2. Jun 16, 2004

### AKG

Make the substitution $\frac{u + at}{c} = \cos \theta$. You should be able to get it down to this:

$$\frac{c}{a^2} \left (u\int _{\frac{\pi}{2}} ^{\arccos \left(\frac{u}{c}\right)} \sin ^2 \theta d\theta - c\int _{\frac{\pi}{2}} ^{\arccos \left(\frac{u}{c}\right)} \sin ^2 \theta \cos \theta d\theta \right )$$

And you can easily solve that on your own.

EDITED to fix limits of integration as per HallsOfIvy's comment.

Last edited: Jun 16, 2004
3. Jun 16, 2004

### HallsofIvy

Staff Emeritus
The substitution might work but the limits of integration are wrong. When t= 0, cos[theta]= u/c so [theta]= cos<sup>-1</sup>(u/c). When t= u/c, cos[theta]= 0 so [theta]= [pi]/2.

4. Jun 17, 2004

### jcsd

Thanks for that, I realized I made a slight error so it became slightly easier to solve.