What is the integral of xln(1+1/(2x)) ?

The integral of xln(1+1/(2x)) is 1/2 x^2ln(1+1/(2x))+1/8 (2x+1-ln|2x+1|)+C

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integral of xln(1+1/(2x))

\int x ln (1 + \frac{1}{2 x}) d x

\frac{1}{2} x^{2} ln (1 + \frac{1}{2 x}) + \frac{1}{8} (2 x + 1 - ln ∣ 2 x + 1 ∣) + C

Solution steps

\int x ln (1 + \frac{1}{2 x}) d x

Apply Integration By Parts

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

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

\int - \frac{x}{2 ( 2 x + 1 )} d x = - \frac{1}{8} (2 x + 1 - ln ∣ 2 x + 1 ∣)

= \frac{1}{2} x^{2} ln (1 + \frac{1}{2 x}) - (- \frac{1}{8} (2 x + 1 - ln ∣ 2 x + 1 ∣))

Simplify

= \frac{1}{2} x^{2} ln (1 + \frac{1}{2 x}) + \frac{1}{8} (2 x + 1 - ln ∣ 2 x + 1 ∣)

Add a constant to the solution

= \frac{1}{2} x^{2} ln (1 + \frac{1}{2 x}) + \frac{1}{8} (2 x + 1 - ln ∣ 2 x + 1 ∣) + C

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What is the integral of xln(1+1/(2x)) ?
The integral of xln(1+1/(2x)) is 1/2 x^2ln(1+1/(2x))+1/8 (2x+1-ln|2x+1|)+C