What is the integral from 0 to ln(x) of (xe^{t^2}) ?

The integral from 0 to ln(x) of (xe^{t^2}) is (sqrt(pi)x\erfi(ln(x)))/2

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integral from 0 to ln(x) of (xe^{t^2})

\int_{0}^{l n (x)} (x e^{t^{2}}) d t

\frac{π x erfi ( ln ( x ) )}{2}

Solution steps

\int_{0}^{l n (x)} x e^{t^{2}} d t

Take the constant out:

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

= x \cdot \int_{0}^{l n (x)} e^{t^{2}} d t

Use the nonelementary integral:

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

= x [\frac{π}{2} erfi (t)]_{0}^{l n (x)}

Compute the boundaries:

\frac{π}{2} erfi (ln (x))

= x \frac{π}{2} erfi (ln (x))

Simplify

= \frac{π x erfi ( ln ( x ) )}{2}

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What is the integral from 0 to ln(x) of (xe^{t^2}) ?
The integral from 0 to ln(x) of (xe^{t^2}) is (sqrt(pi)x\erfi(ln(x)))/2