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Solve the given problems by integration.Find an equation of the curve for which $d y / d x=8 e^{4 x}$ if the curve passes through (0,6).
Calculus 1 / AB
Chapter 28
Methods of Integration
Section 3
The Exponential Form
Integrals
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we're going to find an equation for why, given that the derivative dy dx is equal to eight times each of the four X. And also given that on our curve, Y is equal to whatever Y is equal to. We have this 40.0 comma six. So we'll use that to figure out what C. Is later. So first let's just find the integral of eight E. To the four X. D. X. Which is equal to the integral of Dy so this is just gonna be equal to Y. And so what we can do is we can take out this eight sentence, a constant. So we have eight multiplied by E. To the four x. Dx. And we can also say each of the 4X. That's equal to. Um Well what we can do is we can say U. is equal to four x. And D. U is equal to four. And then we can plug this into our equation. So this would be equal to eight times the integral of E. To the U. Heat that you multiplied by 1/4. Since D. U. Is equal to four. We have to multiply by 1/4 and multiply by D. You. And so now we just have each of the you multiplied by 1/4. You can take the 1/4 out of our integral. So this would be equal to two multiplied by E. To the U. Time's D. U. And the integral of E. To the power of you is just E. To the power of you. So this is equal to two, multiplied by each the power of you plus C. And so we can plug back in what U. is equal to. And it was equal to four X. So this is equal to two times E. To the four X. Plus C. And now what we can do is we can use that point we said we had on our on our function. Why? Which is equal to zero comma six. To figure out what C. Is equal to. So we say since this is equal to why we say, We're sorry. We say six is equal to two, multiplied by E. Race the power of four times zero which is erased, The zeroth power plus C. So then six is equal to two plus C. So then four is equal to see. So now we have our function for why? Which is equal to two E. Reach the Power four x plus four.
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