What is the integral of (x+1)ln(x) ?

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

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integral of (x+1)ln(x)

\int (x + 1) ln (x) d x

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

Solution steps

\int (x + 1) ln (x) d x

Apply Integration By Parts

: ln (x) (\frac{x ^{2}}{2} + x) - \int \frac{1}{2} (x + 2) d x

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

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

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

Simplify

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

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

Add a constant to the solution

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

f (x) = x^{2} x^{3} + 1

\int x \cdot f (x) d x

\frac{d}{d x} (2 y^{2} + 3 x)

\frac{\partial}{\partial y} (x^{3} - 6 x y + 8 y^{3})

\frac{\partial}{\partial x} (cos (7 x + 4 y + z))

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