What is the integral of ln(x)n(x) ?

The integral of ln(x)n(x) is d(xln^2(x)-2xln(x)+2x)+C

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integral of ln(x)n(x)

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

d (x ln^{2} (x) - 2 x ln (x) + 2 x) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

: ln (x) (x ln (x) - x) - \int ln (x) - 1 d x

= d (ln (x) (x ln (x) - x) - \int ln (x) - 1 d x)

\int ln (x) - 1 d x = x ln (x) - 2 x

= d (ln (x) (x ln (x) - x) - (x ln (x) - 2 x))

Expand

ln (x) (x ln (x) - x) - (x ln (x) - 2 x) : x ln^{2} (x) - 2 x ln (x) + 2 x

= d (x ln^{2} (x) - 2 x ln (x) + 2 x)

Add a constant to the solution

= d (x ln^{2} (x) - 2 x ln (x) + 2 x) + C

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What is the integral of ln(x)n(x) ?
The integral of ln(x)n(x) is d(xln^2(x)-2xln(x)+2x)+C