$$\int{e^{x^2} dx}$$
I have this in school many year ago but I forget everything.. and now I don't believed I need it again!
A friend say I need replace exponent with a variable I make so I can make this task easier?
I try like that:
$a = x^2$, $$a' = 2x \Leftrightarrow \frac{da}{dx} = 2x \Leftrightarrow da = 2x \text{ }dx \Leftrightarrow dx = \frac{1}{2x} \text{ } da$$
Now I change all in task with what I write above:
$$\int{e^{x^2} dx} = \int{e^a \cdot \frac{1}{2x} \text{ } da} = \frac{1}{2x}\int{e^a \text{ } da} = \frac{1}{2x} \cdot e^a+c$$
Now put old value in:
$$\int{e^{x^2} dx = \frac{1}{2x} \cdot e^{2x} + c}$$
When I want check solution, online give me the result:
But I don't know this meaning.. There is better method for solve? This is very much time for solving. I think the method I use, if correct, is used for more complicated exponents? Because this exponent is little.
Your specific mistake is in the third line, when you write $\int e^a \frac{1}{2x} da = \frac{1}{2x}\int e^ada$. Because $a$ is a function of $x$, you cannot pull it out. To see that your answer is wrong, try differentiating $\frac{1}{2x}e^{2x} + C$ and observing what you get. It's a theorem that you cannot express $\int e^{x^2} dx$ in terms of elementary functions.