Question

How do you use substitution to integrate `e^ (-5x) dx`?

Answer 1

Here are two solutions:

The "usual way" (as far as my experience), is to turn this into an integral of the form `int e^u du`.

Let `u=-5x`, this makes `du = -5 dx`, so the integral becomes:

`int e^(-5x) dx = -1/5 int e^u du = -1/5 e^u +C = -1/5 e^(-5x) +C`

If you want to be different, turn it into `int u^r du`

We know that `e^(-5x) = (e^x)^-5`.

Alternative Method
If we want to make `u= e^x` so that `du = e^x dx`, we need to change the exponent on `(e^x)^-5`. We write:

`int e^(-5x) dx = int (e^x)^-6 x^x dx`.

Now with the substitution: `u= e^x` so that `du = e^x dx`,

we get

`int e^(-5x) dx = int u^-6 du = (u^(-5))/-5 +C = -1/5 (e^x)^-5 +C`

Which is equal to the answer by the more usual method.