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

The integral of x^2ln(1+x^2) is 1/3 x^3ln(1+x^2)-2/3 ((x^3)/3-x+arctan(x))+C

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integral of x^2ln(1+x^2)

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

\frac{1}{3} x^{3} ln (1 + x^{2}) - \frac{2}{3} (\frac{x ^{3}}{3} - x + arctan (x)) + C

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

= \frac{1}{3} x^{3} ln (1 + x^{2}) - \frac{2}{3} (\frac{x ^{3}}{3} - x + arctan (x)) + C

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What is the integral of x^2ln(1+x^2) ?
The integral of x^2ln(1+x^2) is 1/3 x^3ln(1+x^2)-2/3 ((x^3)/3-x+arctan(x))+C