What is the integral of xln(1+2/(x^2)+1/(x^4)) ?

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

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integral of xln(1+2/(x^2)+1/(x^4))

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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