Help in integration i=1/L∫Vdt || V=(6t+26)/(t^2+10t+21)

[cunhasb]

I hope anyone could give me a hand on this one...

The induced emf in a 5-henry inductor varies as V=(6t+26)/(t^2+10t+21). Recalling that current i=1/L∫Vdt, find the formula for the current i as a function of time t.

This is what I've gotten so far...

∫(6t+26)/(t^2+10t+21)=(6t+26)/(t+3)(t+7)=A/(t+3)+B(t+7)=2/(t+3)+4/(t+7) dt

i=1/5∫2/(t+3)+4/(t+7)dt
i=1/5(ln(t+3)^2+ln(t+7)^4+k...

Well should I stop here since I've gotten only the variable t on the right side of the formula or should differentiate it since they are asking for the formula of currentas a function of time t? If so... is this correct?

di/dt=1/5{[(2t+6)/(t+3)^2]+[(4t+28)/(t+7)^4)]}

Thank you again guys...

 

[Astronuc]

Sorry, I saw this one then lost it, and forgot.

The expression for i(t) seems correct. But one would need and initial condition to solve for an integration constant, or if the limits of integration are t=0 (or t_o and t=t, one would hopefully have i_o, which could be zero if the circuit is open, or there is some constant (DC) current.

This question would be appropriate in electrical engineering or homework sections.

 

[ravenprp]

Hi there,

Your work so far is correct. Since the question is asking for the formula for current as a function of time, you can differentiate the integral expression you have found. This will give you the formula for current as a function of time.

So, continuing from where you left off:

di/dt = 1/5{[(2t+6)/(t+3)^2]+[(4t+28)/(t+7)^4)]}

= 1/5{[2(t+3)/(t+3)^2]+[4(t+7)/(t+7)^4]}

= 1/5{2/(t+3)+4/(t+7)^3}

= 2/(5(t+3))+4/(5(t+7)^3)

= 2/(5t+15)+4/(5t+175)

= (2+4t)/(5(t+3)(t+7)^3)

So, the final formula for current as a function of time is:

i(t) = (2+4t)/(5(t+3)(t+7)^3)

I hope this helps. Good luck with your studies!