Question

Question #85af2

Answer 1

`= 20 e^(0.1 x) +C`

Explanation:

In terms of actually understanding, I find it is easier to look at the reverse process first, namely differentiation. We know for example that, for some constant `alpha`:

`d/dx (e^(alphax) ) = alpha e^(alphax) qquad triangle`

(Incidentally, this follows from the more general conclusion that: `d/dx (e^(f(x)) ) = f'(x) e^(f(x))`,..., which you can reach very quickly by using logarithmic differentiation.)

It follows from integrating `triangle` that:

`color(red) (int d/dx (e^(alpha x) ) \ dx) =int alpha e^(alpha x) \dx`

Or, by realising that in the red term the integration and differentiation "cancel out" (by the Fundamental Theorem of Calculus), and then reversing the order:

`int alpha e^(alpha x) \dx = e^(alpha x) +C`

Or, moving the constant across:
`int e^(alpha x) \dx = 1/alpha e^(alpha x) + C`

Then, adding a new constant `beta`, to reflect your problem:
`int beta e^(alpha x) \dx = beta/alpha e^(alpha x) + C`

So with `alpha = 0.1`, `beta = 2`:

`int 2 e^(0.1 x) \dx = 2/0.1 e^(0.1 x) + C`
`= 20 e^(0.1 x) +C`

That's all rather mechanical and somewhat dry, but I think it might help find a way of doing these that doesn't involve much memory.