What is the integral of 12xln(2x) ?

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

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integral of 12xln(2x)

\int 12 x ln (2 x) d x

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

Solution steps

\int 12 x ln (2 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 (2 x) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

\frac{d}{d x} (e^{x \cdot y})

f (x) = sin (tan (1 + x^{3}))

x \to 3 lim (\frac{x}{2})

\frac{d}{d x} (\frac{1}{arcsec ( x )})

\frac{d}{d t} (cos (wt))

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