What is the integral of x*arctan(x^2) ?

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

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integral of x*arctan(x^2)

\int x \cdot arctan (x^{2}) d x

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

Solution steps

\int x arctan (x^{2}) d x

Apply Integration By Parts

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

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

\int \frac{x ^{3}}{x ^{4} + 1} d x = \frac{1}{4} ln x^{4} + 1

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

Add a constant to the solution

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

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