What is the integral of x^{ln(x)} ?

The integral of x^{ln(x)} is \sqrt[4]{e}(sqrt(pi))/2 \erfi(ln(x)+1/2)+C

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integral of x^{ln(x)}

\int x^{l n (x)} d x

Solution steps

\int x^{l n (x)} d x

Apply u-substitution

= \int e^{u (u + 1)} d u

Expand

u (u + 1) : u^{2} + u

= \int e^{u^{2} + u} d u

Complete the square

u^{2} + u : (u + \frac{1}{2})^{2} - \frac{1}{4}

= \int e^{(u + \frac{1}{2})^{2} - \frac{1}{4}} d u

Apply u-substitution

= \int e^{v^{2} - \frac{1}{4}} d v

Simplify

Take the constant out:

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

Use the nonelementary integral:

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

Substitute back

Add a constant to the solution

What is the integral of x^{ln(x)} ?
The integral of x^{ln(x)} is \sqrt[4]{e}(sqrt(pi))/2 \erfi(ln(x)+1/2)+C