What is the integral of (ln(x)+xln(x)) ?

The integral of (ln(x)+xln(x)) is xln(x)-x+1/2 x^2ln(x)-(x^2)/4+C

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integral of (ln(x)+xln(x))

\int (ln (x) + x ln (x)) d x

x ln (x) - x + \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} + C

Solution steps

\int ln (x) + x ln (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int ln (x) d x + \int x ln (x) d x

\int ln (x) d x = x ln (x) - x

\int x ln (x) d x = \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

= x ln (x) - x + \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

Add a constant to the solution

= x ln (x) - x + \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} + C

What is the integral of (ln(x)+xln(x)) ?
The integral of (ln(x)+xln(x)) is xln(x)-x+1/2 x^2ln(x)-(x^2)/4+C