A Good Way to Solve ∫(-1)ˣdx

Indefinite Integral

Integral calculus operations involving the exponential function are solved using properties of natural logarithms. But in some situations it is necessary to add some transformation of variables, which leads to a resolution technique. This is what we will present in this present text.

The exponential function defined in the form:

and it is otherwise represented as an exponential term whose base corresponds to Euler’s constant, that is,

One can also find situations where it can be defined to any base, that is, if we represent such base by the literal a>0 and belonging to the set of real numbers then we will have:

and so we will have an exponential function in base a.

Another important property is the derivative of the exponential function, that is,

But, for the purpose of our resolution, here we will use a more general syntax, that is, the composite function to determine its antiderivative (definite integral):

where u is a function of x, u(x).

So we have:

or

where C₁ is the constant of integration.

Now we can apply the general result to particularize our question. But the basis of the function requires a positive value. Here we will have to use a trick. Let’s do:

where ɩ is the imaginary number.

Also, using Euler’s formula, we have the identity:

Gathering such information, our integral is in the form, firstly

then we do

and still this

and since u=2x, we now have a result by direct deduction:

It is still possible to simplify, using substitutions:

or

and then

and the end result

Comment

In this exposition, it was initially postponed to articulate the condition that the function of the integrand includes a negative value base, which obliges to support the hypothesis that one is implicitly working on the set of Complex Numbers, ℂ. That is, the problem was subtly stated in a rhetorical way, to be more motivating.