Integration problem. Help! http://imgur.com/GtuBB

Question

anonymous · Answer

My plan was to invert the whole equation and then multiply across by dV

anonymous · Answer

and then using limits, the dt side is fine I'll get t=... but the rest of the equation im not so sure.

unklerhaukus · Answer

separate the variables

unklerhaukus · Answer

$$\frac{\text d V}{\text dt}=\frac{1}{a+bV}$$
$$\Rightarrow\left(a+bV\right)\text dV=\text dt $$
now integrate

anonymous · Answer

Yeah I got that so V(a+bV) = t
So

$$aV+bV^{2}$$=t

anonymous · Answer

I have the solution but its slightly different to what I got.

unklerhaukus · Answer

dont forget to plus c

anonymous · Answer

Apparently it's still wrong! The solution in the text book is $$t=aV+(b/2)V ^{2}$$

unklerhaukus · Answer

$$aV+bV^2=t+c$$
and remember  $V(t)$

$$V(0)=0\qquad$$

unklerhaukus · Answer

oh$$\int bV\text dV=\frac{V^2}{2}$$

anonymous · Answer

Here's what the lecturer wrote down for the answer, it's not very helpful to me.. http://i.imgur.com/Ea8FU.jpg

unklerhaukus · Answer

whops $$=\frac {bV^2}{2}$$

anonymous · Answer

Sorry can you explain that, I still don't why you are dividing by 2??

unklerhaukus · Answer

remember to add one to the index and divide by the new index of the variable being integrated

unklerhaukus · Answer

$$\int\left(a+bV\right)\text dV=\int\text dt$$
$$aV+\frac{bV^2}{2}=t+c$$

anonymous · Answer

Oh you mean like the basic rule of integration x^n+1/n+1

unklerhaukus · Answer

yeah $$\int x\text dx=\frac{x^2}2+c$$

anonymous · Answer

Thanks for the help, I never would have noticed to do that!

unklerhaukus · Answer

did you find the value of c in this question?