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Solve the given problems by integration.For an electric circuit containing a voltage source $E,$ a resistance $R,$ and a capacitance $C,$ an equation relating the charge $q$ on the capacitor and the time $t$ is $q e^{t / R C}=\frac{E}{R} \int e^{t / R C} d t$ See Fig. $28.8 .$ If $q=0$ for $t=0,$ perform the integration and then solve for $q$ as a function of $t$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 3

The Exponential Form

Integrals

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University of Michigan - Ann Arbor

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in this question where electrical circuit Q. Is the church on the capacity and are Here's the resistance. He is the footage sauce and T. Is time we want to find Q. In terms of T. So let's integrate the right hand side. Now before we do that I wanted to fit to the form of integrating it about you respect to you. So my result will be it a value plus see a constant. So let's you be key over RC. So do you over D. T. Will be one over R. C. The you will be one over R. C. E. T. So looks like I need a one over R. C. Over here. So let me do this. E over our. My duty is over here. Let me put one over R. C. Now since what we are seeing is over here. I need to balance by putting RC here in front. Mhm. And my veto power of T over our cities like that remains unchanged. Yeah. We can see that the R. And R. Here cancers. So we will have easy evil power of this is you. And this part over here is the you. Now you can see that you fit the this form. So I'll be using this result. Yeah. So it'll be easy. It's the power you plus let me use K. And another C. You'll be sounds so confusing. Now replace back the you with key over R. C. Because this is a function of T. Okay now to find K. We were so integral. Zero Q go zero. My K. Is going to be minus E. C. So putting it back into over here we will have Q. It'll party over R. C is equals to easy E. T over R C minus easy. Now every time will be divided by this. Every time. Yes, we will get Q is equals to easy minus E. C. e to bomb minus T over R C victor writes out the ec, I'll get one minus either power minus t u r c. That's our answer.

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