It will come in terms of Ei which is a function of complex plane.so needn't worry about its integration.Provided the question you have asked is a^(e^x)

Integration of "a" to the power of "e" power x

To find the integral of "a" to the power of "e" power x, we can use the technique of substitution. This involves substituting a new variable to simplify the expression and make it easier to integrate.

Variable Substitution:
Let's substitute a new variable u, such that u = "e" power x. This substitution will help us simplify the integral.

Differentiation of u:
To find the derivative of u with respect to x, we can use the chain rule. Since u = "e" power x, its derivative will be du/dx = (d/dx)("e" power x) = "e" power x.

Solving for dx:
Rearranging the equation du/dx = "e" power x, we can solve for dx by taking the reciprocal of both sides: dx = (1/"e" power x) du.

Substituting dx and u:
Now, we substitute dx and u in the integral expression. The integral of "a" to the power of "e" power x can be rewritten as:

∫("a" to the power of "e" power x) dx

= ∫("a" to the power of u) (1/"e" power x) du

Simplifying the Integral:
Using the properties of exponents, we can rewrite "a" to the power of u as (e to the power of ln(a)) to the power of u. This simplification allows us to separate the integral into two parts:

= (1/"e" power x) ∫(e to the power of ln(a)) to the power of u du

Integrating the Separated Integral:
The integral of (e to the power of ln(a)) to the power of u with respect to u can be easily evaluated:

= (1/"e" power x) [((e to the power of ln(a)) to the power of u) / ln(a)] + C

= (1/"e" power x) [(a to the power of u) / ln(a)] + C

Final Result:
Therefore, the integral of "a" to the power of "e" power x is equal to (1/"e" power x) [(a to the power of u) / ln(a)] + C, where C is the constant of integration.