If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

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?

  • piceratops ultimate style avatar for user Nusaybah
    At , how is it allowed to 'take a constant out of the integral sign'?
    (8 votes)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user Ajay Kudapa
    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)
    (3 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Matthew Chen
      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.
      (2 votes)
  • spunky sam red style avatar for user Satyam Sharma
    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)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user Ashutosh Randive
    Why exercise of exponential is above this video?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Ahmed Mo Kahen
    why not increase the power of e by 1....
    in the case of 5e^x anti derivative ?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user MuliBoy
      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)
  • blobby green style avatar for user D. Ashley Nelson
    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.
    (0 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user stolenunder
      In general, when you see e, it represents the base of the natural logarithm. Normally most people will never freely use e as a variable (nor should you), since it is normally assumed that e = 2.71828...
      It's just one of those annoying math things that can seem ambiguous, but you are right, you should announce what you're using variables and constant symbols for. But doing so every time is also tedious and I doubt Sal wants to do it every time. Plus I believe he says later something about it being "just a value" so it should be insinuated from that, that he is using e as a constant, not a variable.
      (3 votes)
  • blobby green style avatar for user hashmisyedmuddassir8888
    Intigral of e^x
    (0 votes)
    Default Khan Academy avatar avatar for user

Video transcript

- [Voiceover] We're told that F of seven is equal to 40 plus five, E to the seventh power, and F prime of X is equal to five, E to the X. What is F of zero? So to evaluate F of zero, let's take the anti-derivative of F prime of X, and then we're going to have a constant of integration there, so we can use the information that they gave us up here that F of seven is equal to this. This might look like an expression. Well, it is an expression, but it's really just a number. There's no variables in this, and so we can use that to solve for our constant of integration, and then we will have fully known what F of X is, and we can use that to evaluate F of zero, so let's just do it. So if F prime of X is equal to five, E to the X, then F of X is going to be equal to the anti-derivative of F prime of X, or the anti-derivative of five, E to the X, DX, and this is the thing that I always find amazing about exponentials, and actually, let me just take a step. I'll take that five out of the integral so it becomes a little bit more obvious. And so the anti-derivative of E to the X, well, that's just E to the X because the derivative of E to the X is E to the X, which I find amazing every time I have to manipulate or take the derivative or anti-derivative of E to the X. So this is gonna be five, E to the X, plus C, and you can verify. Take the derivative of five, E to the X, plus C. The derivative of five, E to the X, well, that's five, E to the X, so that works out. Well, and the derivative of C is zero, so you wouldn't see it over here. So now, let's use this information to figure out what C is so that we know exactly what F of X is, and then we can evaluate F of zero. So we know that F of seven, so when X is equal to seven, this expression is going to evaluate to this thing, 40 plus five, E to the seven. So, five times E to the seventh power plus C is equal to 40, plus five, E to the seventh power. And notice, all I did is say, okay, F of seven. Well, if this is F of X-- Let me write this down. So, if this is F of seven, if this is F of X, I just replaced the X with a seven to find F of seven, and we know that F of seven is also going to be equal to that. They gave us that information, but when you just look at this, it's pretty easy to figure out what C is going to be. You can subtract five, E to the seven from both sides, and you see that C is equal to 40. And so we can rewrite F of X. We can say that F of X is equal to five, E to the X, plus C, which is 40. And so now, from that, we can evaluate F of zero. F of zero is going to be five times E to the zero power, plus 40. E to the zero is one, so it's gonna be five times one, which is just five, plus 40, which is equal to 45, and we're done.