Evaluation of the non-elementary integral $\int e^{\lambda x^\alpha} dx, \alpha\ge2$, and other related integrals

A formula for the non-elementary integral $\int e^{\lambda x^\alpha} dx$ where $\alpha$ is real and greater or equal two, is obtained in terms of the confluent hypergeometric function $_1F_1$. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to $\alpha = 2$, using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function $_1F_1$ and another one in terms of the hypergeometric function $_1F_2$, are obtained for each of these integrals, $\int \cosh(\lambda x^\alpha)dx$, $\int \sinh(\lambda x^\alpha)dx$, $\int \cos(\lambda x^\alpha)dx$ and $\int \sin(\lambda x^\alpha)dx$, $\lambda\in \mathbb{C}, \alpha\ge2$. And the hypergeometric function $_1F_2$ is expressed in terms of the confluent hypergeometric function $_1F_1$. Some of the applications of the non-elementary integral $\int e^{\lambda x^\alpha}dx,\alpha\ge2$ such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.


Introduction
Definition 1. An elementary function is a function of one variable built up using that variable and constants, together with a finite number of repeated algebraic operations and the taking of exponentials and logarithms [6].
In 1835, Joseph Liouville established conditions in his theorem, known as Liouville 1835's Theorem [4,6], which can be used to determine whether an indefinite integral is elementary or nonelementary. Using Liouville 1835's Theorem, one can show that the indefinite integral e λx α dx, α ≥ 2, is non-elementary [4], and to my knowledge, no one has evaluated this non-elementary integral before.
For instance, if α = 2, λ = −β 2 < 0, where β is a real constant, the area under the Gaussian Bell curve can be calculated using double integration and then polar coordinates to obtain (1. 2) The Central limit Theorem (CLT) in Probability theory [2] states that the probability that a random variable x does not exceed some observed value z 2 Victor Nijimbere So if we know the antiderivative of the function g(x) = e λx 2 , we may choose to use the FTC to calculate the cumulative probability P (X < z) in (1.3) when the value of z is given or is known, rather than using numerical integration.
The Maxwell-Boltsman distribution in gas dynamics, where θ and γ are some positive constants that depend on the properties of the gas and v is the gas speed, is another application.
There are many other examples where the antiderivative of g(x) = e λx α , α ≥ 2 can be useful. For example, using the FTC, formulas for integrals such as where n is a positive integer, can be obtained if the antiderivative of g(x) = e λx α , α ≥ 2 is known. In this paper, the antiderivative of g(x) = e λx α , α ≥ 2, is expressed in terms of a special function, the confluent hypergeometric 1 F 1 [1]. And the confluent hypergeometric 1 F 1 is an entire function [3], and its properties are well known [1,5]. The main goal here is to consider the most general case with λ complex (λ ∈ C), evaluate the non-elementary integral e λx α , α ≥ 2 and thus make possible the use of the FTC to compute the definite integral are evaluated in terms of 1 F 1 for any constant λ. They are also expressed in terms of the hypergeometric 1 F 2 . And some expressions of the hypergeometric function 1 F 2 in terms of the confluent hypergeometric function 1 F 1 are therefore obtained. For reference, we shall first define the confluent confluent hypergeometric function 1 F 1 and the hypergeometric function 1 F 2 before we proceed to the main aims of this paper (see sections 2 and 3).

Evaluation of
, λ is an arbitrarily constant and α ≥ 2, is the antiderivative of the function g(x) = e λx α . Thus, P r o o f. We expand g(x) = e λx α as a Taylor series and integrate the series term by term. We also use the Pochhammer's notation [1] for the gamma function, Γ(a + n) = Γ(a)(a) n , where (a) n = a(a + 1) · · · (a + n − 1), and the property of the gamma function Γ(a + 1) = aΓ(a) [1]. For example, Γ(n + a + 1) = (n + a)Γ(n + a). We then obtain (2.2) Example 1. We can now evaluate x 2n e λx 2 dx in terms of the confluent hypergeometric function. Using integration by parts, 1. For instance, for n = 1, 2. For n = 2, Victor Nijimbere Example 2. Using the method of integrating factor, the first-order ordinary differential equation has solution Assuming that the function G(x) (see Proposition 1) is unknown, in the following lemma, we use the properties of function g(x) to establish the properties of G(x) such as the inflection points and the behavior as x → ±∞.
1. If the real part of λ is negative (< 0) and α is even, then the limits lim x→−∞ G(x) and lim x→+∞ G(x) are finite (constants). And thus the Lebesgue integral 3. And if λ ∈ R and λ < 0, and α is even, then the limits lim x→−∞ G(x) and lim x→+∞ G(x) are finite. And there exists real constant θ > 0 such that limits 1. For complex λ = λ r + iλ i , where the subscript r and i stand for real and imaginary parts respectively, the function g(x) = g(z) = e z α where z = (λ r + iλ i ) 1/α x, α ≥ 2, is an entire function on C. And if λ r < 0 and α is even implies Re(z α ) is always negative regardless of the values of x. And so, if |z| → ∞ (or x → ±∞), then g(z) = 0 (g(z) → 0) (or g(x) = 0 as x → ±∞). Therefore by Liouville theorem, G(z) has to be constant as |z| → ∞, and so is For λ r < 0 and α odd, the limit lim x→−∞ e λrx α diverges and so does the integral ∞ −∞ e λrx α dx. Therefore, the Lebesgue integral ∞ −∞ |e λx α |dx has to diverge too. On the other hand, for λ r > 0, the limit lim x→+∞ e λrx α diverges, and so does the integral ∞ −∞ e λr x α dx regardless of the value of α. Therefore, the Lebesgue integral 3. For λ = λ r (λ ∈ R), both g(x) and G(x) are analytic on R. Using this fact and the fact that for even α and λ r < 0, ∞ −∞ |e λx α |dx < ∞ implies that for even α and λ r < 0, G(x) has to be constant as x → ±∞. In addition, the fact that G ′′ (x) < 0 if x < 0 and G ′′ (x) > 0 if x > 0 implies that, G(x) is concave upward on the interval (∞, 0) while is concave downward on the interval (0, +∞). Moreover, the fact that g(x) = G ′ (x) is symmetric about the y-axis (even) implies that G(x) has to be antisymmetric about the y-axis (odd). Hence there exists a real positive constant θ > 0 such that limits lim x→−∞ G(x)= − θ and lim x→+∞ G(x)=θ.
Its graph as a function of x, sketched using MATLAB, is shown in Figure 1. It is in agreement with Lemma 1. It is actually seen in Figure 1 that (0, 0) is an inflection point and that G(x) reaches some constants as x → ±∞ as predicted by Lemma 1.
In the following lemma, we obtain the values of G(x), the antiderivative of the function g(x) = e λx α , as x → ±∞ using the asymptotic expansion of the confluent hypergeometric function 1 F 1 .
1. Then for |x| ≫ 1, 2. Let α ≥ 2 and be even, and let λ = −β 2 , where β is a real number, preferably positive. Then and 3. And by the FTC, Victor Nijimbere P r o o f.
Example 4. Setting λ = −β 2 = −1, β = 1 and α = 2 in Lemma 2 gives and (2.29) This implies θ = √ π/2 in Lemma 1. And this is exactly the value of G(x) as x → ∞ in Figure 1. We also have lim x→−∞ G(x) = −θ = − √ π/2 as in Figure 1. Using the FTC, we readily obtain (2.32) Example 5. In this example, the integral where n is a positive integer, is evaluated using Proposition 1 and the asymptotic expression (2.9). Setting λ = 1 and α = 2n + 1 in Proposition 1 , and using (2.9) gives The non-elementary integral e λx α dx, α ≥ 2, and other related integrals 9 Theorem 1. For any A and B, the FTC gives where G is the antiderivative of the function g(x) = e λx α and is given in Proposition 1. And λ is any complex or real constant, and α ≥ 2.
where λ is any constant, is the antiderivative of g(x) = e λx α , α ≥ 2 by Proposition 1, Lemma 1 and Lemma 2. And since the FTC works for A = −∞ and B = 0 in (2.30), A = 0 and B = +∞ in (2.31) and A = −∞ and B = +∞ in (2.32) by Lemma 2 if λ = 1 and α = 2, and for all λ < 0 and all even α ≥ 2, then it has to work for other values of A, B ∈ R and for any λ ∈ C and α ≥ 2. This completes the proof.
Example 7. Using integration by parts and applying Theorem 1, the Maxwell-Bortsman distribution is written in terms of the confluent hypergeometric 1 F 1 as (2.39)

Other related non-elementary integrals
, where 1 F 2 is a hypergeometric function [1], λ is an arbitrarily constant and α ≥ 2, is the antiderivative of the function g(x) = cosh (λx α ). Thus, P r o o f. We proceed as before. We expand g(x) = cosh (λx α ) as a Taylor series and integrate the series term by term, use the Pochhammers notation [1] for the gamma function, Γ(a + n) = Γ(a)(a) n , where (a) n = a(a + 1) · · · (a + n − 1), and the property of the gamma function Γ(a + 1) = aΓ(a) [1]. We also use the Gamma duplication formula [1]. We then obtain Proposition 3. The function where 1 F 2 is a hypergeometric function [1], λ is an arbitrarily constant and α ≥ 2, is the antiderivative of the function g(x) = sinh (λx α ). Thus, P r o o f. As above, we expand g(x) = sinh (λx α ) as a Taylor series and integrate the series term by term, use the Pochhammers notation [1] for the gamma function, Γ(a + n) = Γ(a)(a) n , where (a) n = a(a + 1) · · · (a + n − 1), and the property of the gamma function Γ(a + 1) = aΓ(a) [1]. We also use the Gamma duplication formula [1]. We then obtain We also can show as above that Theorem 2. For any constants α and λ, Hence, comparing (3.1) with (3.9) gives (3.7). Using Proposition 1, on the other hand, we obtain Hence, comparing (3.5) with (3.10) gives (3.8).

Conclusion
The non-elementary integral e λx α dx, where λ is an arbitrary constant and α ≥ 2, was expressed in term of the confluent hypergeometric function 1 F 1 . And using the properties of the confluent hypergeometric function 1 F 1 , the asymptotic expression for |x| ≫ 1 of this integral was derived too. As established in Theorem 1, the definite integral (1.6) can now be computed using the FTC. For example, one can evaluate the area under the Gaussian Bell curve using the FTC rather than using double integration and then polar coordinates. One can also choose to use Theorem 1 to compute the cumulative probability for the normal distribution or that for the Maxwell-Bortsman distribution as shown in examples 2 and 2.
On one hand, the integrals cosh(λx α )dx, sinh(λx α )dx, cos(λx α )dx and sin(λx α )dx, α ≥ 2, were evaluated in terms of the confluent hypergeometric function 1 F 1 , while on another hand, they were expressed in terms of the hypergeometric 1 F 2 . This allowed to express the hypergeometric function 1 F 2 in terms of the confluent hypergeometric function 1 F 1 (Theorems 2 and 3).