Question

How do you find the integral of $e^{x^2}$ ?

Answer 1

Hint: In the above question we have to find the integral of $e^{x^2}$ . As you know that the integration of an exponential function is the same. Moreover, in integration, we cannot integrate any constant number. So in the final answer, we add a constant. So let us see how we can solve this problem.

Step by step solution:
Let us find the integral of $e^{x^2}$ . This question cannot be solved or we can say that it has no finite solution. We will use an infinite series to solve this problem.
 $\Rightarrow e^x=1+x+{\displaystyle \frac{x^2}{2!}}+{\displaystyle \frac{x^3}{3!}}...=1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{3}}+...$ (for all x), it follows that,
 $\Rightarrow e^{x^2}=1+x^2+{\displaystyle \frac{x^4}{2}}+{\displaystyle \frac{x^6}{6}}+...$ (for all x)
We will use the formula of $\int x^n={\displaystyle \frac{x^{n+1}}{n+1}}$ and we will integrate each of them.
 $\int 1=x,\int x^2={\displaystyle \frac{x^{2+1}}{2+1}},\int {\displaystyle \frac{x^4}{2}}={\displaystyle \frac{x^{4+1}}{2(4+1)}},\int {\displaystyle \frac{x^6}{6}}={\displaystyle \frac{x^{6+1}}{6(6+1)}}$
Applying integration on both the sides
 $\Rightarrow \int e^{x^2}=\int (1+x^2+{\displaystyle \frac{x^4}{2}}+{\displaystyle \frac{x^6}{6}}+...)dx$
 $=C+x+{\displaystyle \frac{x^3}{3}}+{\displaystyle \frac{x^5}{10}}+{\displaystyle \frac{x^7}{42}}+...$
We can see that we don’t get any feasible finite solution for the given problem.

Note:
In the above solution we solved the integration for $e^{x^2}$ but we get an infinite solution. According to Wolfram Alpha theorem, the antiderivative whose graph goes through the origin as ${\displaystyle \frac{\sqrt{\pi }}{2}}erfi(x)$ , where erfi(x) is called the "imaginary error function". Also, we have given the constant term in the integration as C.