Question

What is the integral of \[{e^{{x^3}}}\]?

Answer 1

Hint: In order to determine the integral of the given exponential function. In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration.

Complete step by step solution:
We are given the exponential function is \[\int {{e^{{x^3}}}dx} \]
Now, Let us consider, \[z = {x^3}\]
Differentiating the exponential o ‘z’ with respect to ‘x’ , then
\[
   \Rightarrow 3{x^2}dx = dz \\
   \Rightarrow dx = \dfrac{{dz}}{{3{x^2}}} \\
 \]
Comparing the exponential function \[z = {x^n}\] with \[dx = \dfrac{1}{n}{z^{\dfrac{1}{n} - 1}}dz\], so we can write
\[
  dx = \dfrac{1}{3}{z^{1 - 3}}dz \\
  dx = \dfrac{1}{3}{z^{\dfrac{1}{3} - 1}}dz \\
 \]
Since, \[n = 3\]
Now we can substitute the ‘dx’ and ‘z’ value into the given equation
\[\int {{e^{{x^3}}}dx} = \int {{e^z}\dfrac{{dz}}{{3{x^2}}}} \]\[\int {{e^{{x^3}}}dx} = \int {{e^z}\dfrac{{dz}}{{3{x^2}}}} \]
We take the integral limit as \[0\]to \[\infty \], we get
\[\int {{e^{{x^3}}}dx} = \int {{e^z}\dfrac{1}{3}{z^{\dfrac{1}{3} - 1}}dz} \]
\[\int {{e^{{x^3}}}dx} = \int\limits_0^\infty {{e^z}\dfrac{1}{3}{z^{\dfrac{1}{3} - 1}}dz} \]
Expand the integral values on the exponential function, we can get
\[\int {{e^{{x^3}}}dx} = \dfrac{1}{3}\left( {\int\limits_0^\infty {{e^z}{z^{\dfrac{1}{3} - 1}}dz - \int\limits_z^\infty {{e^z}{z^{\dfrac{1}{3} - 1}}dz} } } \right) + c\]
On compare the formula for indefinite integral \[\Gamma (n,z) + d\] with the above derivative equation, the
\[\int {{e^{{x^3}}}dx} = \dfrac{1}{3}\Gamma (\dfrac{1}{3},{x^3}) + d\]
Where $d$ and $c$ are constant.
Hence, the integral of \[{e^{{x^3}}}\]is \[\dfrac{1}{3}\Gamma (\dfrac{1}{3},{x^3}) + d\].

Additional information:
In integral, there are two types of integrals in maths:
> Definite Integral
> Indefinite Integral
Definite Integral:
An integral that contains the upper and lower limits then it is a definite integral. On a real line, x is restricted to lie. Riemann Integral is the other name of the Definite Integral.
A definite Integral is represented as:
\[\int\limits_a^b {f(x)dx} \] Indefinite Integral:
Indefinite integrals are defined without upper and lower limits. It is represented as:
\[\smallint f(x)dx = F(x) + C\]
Where C is any constant and the function \[f\left( x \right)\] is called the integrand.

Note:
We can derive the exponential function \[{x^n}\] as follows
Let the \[z = {x^n}\]
Differentiate with respect to x
\[
  dz = n{x^{n - 1}}dx \\
  dx = \dfrac{1}{{n{z^{n - 1}}}}dz \\
 \]
We can change the denominator function as a numerator. So, it changed to negative exponential.
\[
  dx = \dfrac{1}{n}{z^{ - (1 - n)}}dz \\
  dx = \dfrac{1}{n}{z^{n - 1}}dz \Rightarrow \dfrac{1}{n}{z^{\dfrac{1}{n} - 1}}dz \\
 \]