Question

What is the integral of `e^(3x)`?

Answer 1

The answer is `inte^(3x)dx=e^(3x)/3`.

So we have `f(x) = e^(3x) = g(h(x))`, where `g(x) = e^x` and `h(x) = 3x`.

The antiderivative of such a form is given by :

`intg(h(x))*h'(x)dx = G(h(x))`

We know that the derivative of `h(x) = 3x` is `h'(x)=3`.

We also know that the antiderivative of `g(x) = e^x` is `G(x) = e^x`.

We have `inte^(3x)dx` but, with our formula, we can only calculate `inte^(3x)*3dx`, so what we will do is :

`inte^(3x)dx = 1/3inte^(3x)*3dx = e^(3x)/3`.

That's it.