What is the integral of 2(ln(x)+1)e^{xln(x)} ?

The integral of 2(ln(x)+1)e^{xln(x)} is 2x^x+C

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integral of 2(ln(x)+1)e^{xln(x)}

\int 2 (ln (x) + 1) e^{x l n (x)} d x

2 x^{x} + C

Solution steps

\int 2 (ln (x) + 1) e^{x l n (x)} d x

Take the constant out:

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

= 2 \cdot \int (ln (x) + 1) e^{x l n (x)} d x

Apply u-substitution

: \int e^{u} d u

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

Use the common integral:

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

= 2 e^{u}

Substitute back

u = x ln (x)

= 2 e^{x l n (x)}

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

= 2 x^{x}

Add a constant to the solution

= 2 x^{x} + C

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What is the integral of 2(ln(x)+1)e^{xln(x)} ?
The integral of 2(ln(x)+1)e^{xln(x)} is 2x^x+C