A ( presumably ) basic definite integral problem

[Ognjen]

I came across a basic integration problem in physics ( kinematics ) and I am baffled with the solution given in the textbook ( it obviously skipped a few steps deeming them ''too obvious'' to explicitly address ). Here goes:

$$S(t) = 2\sqrt{2}*v_s \int^0_1 \frac{dt}{R_0*\sqrt{1 - \frac{4v_st}{(R_0)^2}}}$$
v_s and R_0 are constants ( sector velocity and vector projection on the radial axis in a polar coordinate system, respectively ). I don't think the physics of this matters though, which is why I'm posting this on a math forum.

According to the solution, the integral resolves to:

$$-R_0\sqrt{2} * (\sqrt{1-\frac{4v_st}{(R_0)^2}})\Biggr|_{0}^{1}$$
How does the integral resolve to this ? I tried solving it with substitution ( aiming for dx\(1-x^2) ), but inverse trig functions are nowhere to be found in their solution, so I assume they didn't walk this route at all.

I would appreciate any help !

 

[Deleted member 4993]

I came across a basic integration problem in physics ( kinematics ) and I am baffled with the solution given in the textbook ( it obviously skipped a few steps deeming them ''too obvious'' to explicitly address ). Here goes:

$$S(t) = 2\sqrt{2}*v_s \int^0_1 \frac{dt}{R_0*\sqrt{1 - \frac{4v_st}{(R_0)^2}}}$$
v_s and R_0 are constants ( sector velocity and vector projection on the radial axis in a polar coordinate system, respectively ). I don't think the physics of this matters though, which is why I'm posting this on a math forum.

According to the solution, the integral resolves to:

$$-R_0\sqrt{2} * (\sqrt{1-\frac{4v_st}{(R_0)^2}})\Biggr|_{0}^{1}$$
How does the integral resolve to this ? I tried solving it with substitution ( aiming for dx\(1-x^2) ), but inverse trig functions are nowhere to be found in their solution, so I assume they didn't walk this route at all.

I would appreciate any help !

Substitute:

x = 1 - $\displaystyle \frac{4v_s}{(R_0)^2}$ * t $\displaystyle \ \to \ \ x = 1 - A*t$

 

[Ognjen]

Substitute:

x = 1 - $\displaystyle \frac{4v_s}{(R_0)^2}$ * t $\displaystyle \ \to \ \ x = 1 - A*t$

Thank you immensely !