What is the integral of 12xln(3x) ?

The integral of 12xln(3x) is 12(1/2 x^2ln(3x)-(x^2)/4)+C

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integral of 12xln(3x)

\int 12 x ln (3 x) d x

12 (\frac{1}{2} x^{2} ln (3 x) - \frac{x ^{2}}{4}) + C

Solution steps

\int 12 x ln (3 x) d x

Take the constant out:

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

= 12 \cdot \int x ln (3 x) d x

Apply Integration By Parts

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

= 12 (\frac{1}{2} x^{2} ln (3 x) - \int \frac{x}{2} d x)

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= 12 (\frac{1}{2} x^{2} ln (3 x) - \frac{x ^{2}}{4})

Add a constant to the solution

= 12 (\frac{1}{2} x^{2} ln (3 x) - \frac{x ^{2}}{4}) + C

\frac{d y}{d t} = t y^{\frac{1}{3}}

2 + e^{2 t} + e^{- 2 t}

\frac{\partial}{\partial x} (36 (\frac{x}{x + y}) - x)

(x^{2} + 1) (\frac{d y}{d x}) + 8 x (y - 1) = 0, y (0) = 6

\frac{\partial}{\partial x} (2 x^{3} y + 1)

What is the integral of 12xln(3x) ?
The integral of 12xln(3x) is 12(1/2 x^2ln(3x)-(x^2)/4)+C