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

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

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integral of xln(1+x^4)

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

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

Solution steps

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

Apply Integration By Parts

= \frac{1}{2} x^{2} ln (1 + x^{4}) - \int \frac{2 x ^{5}}{1 + x ^{4}} d x

\int \frac{2 x ^{5}}{1 + x ^{4}} d x = x^{2} - arctan (x^{2})

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

Simplify

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

Add a constant to the solution

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

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