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

The integral of xln(2x^2+1) is 1/2 x^2ln(2x^2+1)-2((x^2)/4-1/8 ln|2x^2+1|)+C

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integral of xln(2x^2+1)

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

\frac{1}{2} x^{2} ln (2 x^{2} + 1) - 2 (\frac{x ^{2}}{4} - \frac{1}{8} ln 2 x^{2} + 1) + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{2} x^{2} ln (2 x^{2} + 1) - \int \frac{2 x ^{3}}{2 x ^{2} + 1} d x

= \frac{1}{2} x^{2} ln (2 x^{2} + 1) - \int \frac{2 x ^{3}}{2 x ^{2} + 1} d x

\int \frac{2 x ^{3}}{2 x ^{2} + 1} d x = 2 (\frac{x ^{2}}{4} - \frac{1}{8} ln 2 x^{2} + 1)

= \frac{1}{2} x^{2} ln (2 x^{2} + 1) - 2 (\frac{x ^{2}}{4} - \frac{1}{8} ln 2 x^{2} + 1)

Add a constant to the solution

= \frac{1}{2} x^{2} ln (2 x^{2} + 1) - 2 (\frac{x ^{2}}{4} - \frac{1}{8} ln 2 x^{2} + 1) + C

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What is the integral of xln(2x^2+1) ?
The integral of xln(2x^2+1) is 1/2 x^2ln(2x^2+1)-2((x^2)/4-1/8 ln|2x^2+1|)+C