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

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

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integral of ln(2x)(x)

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

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

Solution steps

\int ln (2 x) 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

4 x^{4} - 4 x^{2}, 6 x^{2}

x \to 0 + lim (\frac{1}{x} sin (\frac{1}{x}))

\int tan^{2} (x) d x

f (x) = (1 + x)^{- \frac{6}{5}}

x \to 3 lim (\frac{5 \cdot 4 ^{x}}{9 ^{x}})

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