What is the integral of (ln(x)+1)e^{xln(x)} ?

The integral of (ln(x)+1)e^{xln(x)} is x^x+C

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integral of (ln(x)+1)e^{xln(x)}

\int (ln (x) + 1) e^{x l n (x)} d x

x^{x} + C

Solution steps

\int (ln (x) + 1) e^{x l n (x)} d x

Apply u-substitution

: \int e^{u} d u

= \int e^{u} d u

Use the common integral:

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

= e^{u}

Substitute back

u = x ln (x)

= e^{x l n (x)}

Simplify

e^{x l n (x)} : x^{x}

= x^{x}

Add a constant to the solution

= x^{x} + C

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What is the integral of (ln(x)+1)e^{xln(x)} ?
The integral of (ln(x)+1)e^{xln(x)} is x^x+C