What is the integral of (x+3)ln(x) ?

The integral of (x+3)ln(x) is ln(x)((x^2)/2+3x)-1/2 ((x^2)/2+6x)+C

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integral of (x+3)ln(x)

\int (x + 3) ln (x) d x

ln (x) (\frac{x ^{2}}{2} + 3 x) - \frac{1}{2} (\frac{x ^{2}}{2} + 6 x) + C

Solution steps

\int (x + 3) ln (x) d x

Apply Integration By Parts

: ln (x) (\frac{x ^{2}}{2} + 3 x) - \int \frac{1}{2} (x + 6) d x

= ln (x) (\frac{x ^{2}}{2} + 3 x) - \int \frac{1}{2} (x + 6) d x

\int \frac{1}{2} (x + 6) d x = \frac{1}{2} (\frac{x ^{2}}{2} + 6 x)

= ln (x) (\frac{x ^{2}}{2} + 3 x) - \frac{1}{2} (\frac{x ^{2}}{2} + 6 x)

Add a constant to the solution

= ln (x) (\frac{x ^{2}}{2} + 3 x) - \frac{1}{2} (\frac{x ^{2}}{2} + 6 x) + C

\frac{d}{d x} (x^{4} y^{3})

\int sin (π) t d t

x^{2} y \cdot \frac{d y}{d x} - 9 = 0

x^{^{''}} - x^{^{'}} + x = 0

x = y^{2} - 5, x = 3 - y^{2}

What is the integral of (x+3)ln(x) ?
The integral of (x+3)ln(x) is ln(x)((x^2)/2+3x)-1/2 ((x^2)/2+6x)+C