Help with physics/maths

[Yami Silver]

Hello NB, I'm studying A level Further Maths and I'm currently doing the applied unit M2 (M standing for mechanics).
So, I have a simple question: How would one integrate/differentiate ae^(kt)? I missed the lesson on that and I won't be able to see my teacher till Tuesday and I'm practicing past exam papers.
Thank you

 

[Ken Kaneki]

A= const
K= const ?

d(ae^(kt))/dt =a* d(e^kt)/dt = ake^(KT)

 

[Amir el genio]

∫e^(-kt) dt = ∫(e^u)(-du/k) = ∫(e^u)(-1/k)du
= (-1/k)∫e^u du [since k is a constant, we can pull that factor outside the integral]
= (-1/k)e^u + C [the integral of e^x dx is e^x]

Now reverse the earlier substitution:
= (-1/k)e^(-kt)+C

Which can also be written as:
-1/[ke^(kt)] + C

 

[Yami Silver]

A= const
K= const ?

d(ae^(kt))/dt =a* d(e^kt)/dt = ake^(KT)

Forgot to mention - yeah, k and a are constants, do not mistake a for acceleration, probably should have used a different letter lol