What is the integral of e^{3t^2} ?

The integral of e^{3t^2} is (sqrt(pi))/(2sqrt(3))\erfi(sqrt(3)t)+C

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integral of e^{3t^2}

\int e^{3 t^{2}} d t

\frac{π}{2 3} erfi (3 t) + C

Solution steps

\int e^{3 t^{2}} d t

= \int e^{(3 t)^{2}} d t

Apply u-substitution

: \int e^{u^{2}} \frac{1}{3} d u

= \int e^{u^{2}} \frac{1}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int e^{u^{2}} d u

Use the nonelementary integral:

\int e^{u^{2}} d u = \frac{π}{2} erfi (u)

= \frac{1}{3} \cdot \frac{π}{2} erfi (u)

Substitute back

u = 3 t

= \frac{1}{3} \cdot \frac{π}{2} erfi (3 t)

Simplify

\frac{1}{3} \cdot \frac{π}{2} erfi (3 t) : \frac{π}{2 3} erfi (3 t)

= \frac{π}{2 3} erfi (3 t)

Add a constant to the solution

= \frac{π}{2 3} erfi (3 t) + C

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What is the integral of e^{3t^2} ?
The integral of e^{3t^2} is (sqrt(pi))/(2sqrt(3))\erfi(sqrt(3)t)+C