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## AP®︎/College Calculus AB

### Unit 7: Lesson 6

Finding particular solutions using initial conditions and separation of variables

# Particular solutions to differential equations: exponential function

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.E (LO)
,
FUN‑7.E.1 (EK)
,
FUN‑7.E.2 (EK)
,
FUN‑7.E.3 (EK)
Sal finds f(0) given that f'(x)=5eˣ and f(7)=40+5e⁷.

## Want to join the conversation?

• At , how is it allowed to 'take a constant out of the integral sign'?
• Since integral is in a way an infinite sum, 5 is constant with each term (it doesn't change) so you can just factor it out.
• What if you want to integrate an equation that has the variable in its exponent, and is also being multiplied with a constant?
Example:Integration of (0.67*e^0.044x)
• Well, you can take 0.67 out of the integral, then you can use u-substitution for e^0.044x, setting u =
0.044x. Then du = 0.044dx, or dx = du/0.044 = (1/0.044)du. Then the integral becomes 0.67∫(e^u)*(1/0.044)du. You can take 1/0.044 out of the integral since it is a constant. The integral of e^u is e^u. But you need to unsubstitute the u, so the answer is (0.67/0.044)*e^0.044x, or 15.227e^0.044x. Try taking the derivative of this to double check! Hope this made sense.
• This is a bit of an off topic question, but I am learning integral calculus to comprehensively understand the Basel problem. So what should I target in order to comprehend the famous problem quicker? Thanks
(1 vote)
• Why exercise of exponential is above this video?
(1 vote)
• why not increase the power of e by 1....
in the case of 5e^x anti derivative ?
(1 vote)
• Because e^x is an exponential function, and the power rule is not enough to evaluate its derivative. The power rule is applicable only to some variable in the base of a power and a constant in the exponent of that power. But with e^x the variable is in the exponent, and e (the base) is just constant number!
So we use the chain rule to differentiate e^x. If you're still not sure why, watch the lessons about the power rule and derivative of e^x
(1 vote)
• As Sal does not extract a value from the function f(7)= 40 + 5e^7, I can safely assume e is a variable and not the number e. We have two variables, x and e, and I am assuming a domain of all real numbers. At , Sal says f(0)=5e^0 + 40 = 5(1) +40=45. Sal assumes e to be positive but he no where stated in defining his problem "for all e>0." Likewise, if e is a function of x and x's domain is all real numbers, the range is not inherently limited to positive values for e. My argument is that the answer is undefined as e^0 could be -1 or +1.
• The integral of `e^x` is just that- `e^x`.