Question

What is the antiderivative of e^x^2?

Answer 1

Answer: The antiderivative of
`e^{x^(2)}` is
`√(\pi) \sqrt{e^(x^2)-1} +C`.

Step-by-step explanation:

`I=\int{e^(x^2)dx}\\I^2=\int{e^(x^2)dx}\int{e^(x^2)dx}`

Using dawson integral.

`I^2=\int{e^(x^2)dx}\int{e^(y^2)dy}`

`I^2=\int{\int{e^(x^2+y^2)dxdy}}`

Put
`x^2+y^2=r^2`

`I^2=\int{\int{e^(r^2)rdrd\theta}}`

`\int_(0)^(2\pi){d\theta}\int_(0)^(x){re^(r^2)dr}`

Use substitution method and sunbstitute
`r^2=t`.

`\int_(0)^(2\pi){d\theta}\int_(0)^(x^2){(1)/(2)e^tdt}`

`I^2=(1)/(2)|\theta|_(0)^(2\pi)|e^t|_(0)^(x^2)\\I^2=(1)/(2)(2\pi)(e^(x^2)-1)\\I=√(\pi)\sqrt{e^(x^2)-1}`.

Therefore, the antiderivative of
`e^{x^(2)}` is
`√(\pi) \sqrt{e^(x^2)-1} +C`.