Given $v(t)=2\sin(e^t/4)+1$ how do i find the average velocity of the particle for the time period $0-6$.

Question

I'm also given acceleration and $x(0)=2$ but i'm not sure any of that information will be beneficial helping to solve this problem.

I believe I have to first integrate the equation for velocity so I can use the equation: s(b)-s(a)/b-a, but i'm not sure how to go about this.

Answer 1

I know that this is not a real answer but I don't think that function can be integrated! I may be wrong thou'.

Answer 2

With the substitution $u = e^t$, the problem becomes $$1 + {1\over 3}\int_{{1/4}}^{\exp{\!(6)}/4}{{\sin{u}\over u}\,du}\,.$$

The integral $\int_0^z{{\sin{u}\over u}\,du}$ defines the so-called Sine Integral function, denoted ${\rm Si}(z)$. So the answer to your problem is expressible as $$1 + {1\over 3}\left[{\rm Si}\left({e^6\over 4}\right) - {\rm Si}\left({1\over 4}\right)\right] \approx 1.437\,.$$

Alternatively, you could use technology to integrate numerically from the beginning.