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

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

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integral of xln(x+1/x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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