What is the integral of x*ln(2x) ?

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

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integral of x*ln(2x)

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

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

Solution steps

\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

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

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

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

Add a constant to the solution

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

y^{^{'}} = y (1 - 0.0004 y)

x = 4 y^{2}, x = 8 + 2 y^{2}

\frac{d}{d x} (3 x^{2} e^{x^{3}})

t^{2} y^{^{'}} + 2 t y = 3

\int \frac{3}{x ( x ^{2} + 25 )} d x

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