What is the integral of (2x+3)ln(x) ?

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

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integral of (2x+3)ln(x)

\int (2 x + 3) ln (x) d x

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

Solution steps

\int (2 x + 3) ln (x) d x

Apply Integration By Parts

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

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

\int x + 3 d x = \frac{x ^{2}}{2} + 3 x

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

Simplify

ln (x) (x^{2} + 3 x) - (\frac{x ^{2}}{2} + 3 x) : x^{2} ln (x) + 3 x ln (x) - \frac{x ^{2}}{2} - 3 x

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

Add a constant to the solution

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

13 x e^{x}

x \to 0 lim (14 x + 7 h)

\int 2 x arctan (x) d x

\frac{d y}{d x} = \frac{x}{10 y}

y = 2 x^{3} - x^{2} + 1, (2, 13)

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