Integral Solve for u + at^2/c^2

[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

 

[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.

 

[HallsofIvy]

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

 

[jcsd]

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