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

The integral of x*(arctan(x))^2 is 1/2 x^2arctan^2(x)-xarctan(x)+1/2 arctan^2(x)+ln|sqrt(x^2+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^{2} (x) - x arctan (x) + \frac{1}{2} arctan^{2} (x) + ln x^{2} + 1 + C

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

= \frac{1}{2} x^{2} arctan^{2} (x) - x arctan (x) + \frac{1}{2} arctan^{2} (x) + ln x^{2} + 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^2(x)-xarctan(x)+1/2 arctan^2(x)+ln|sqrt(x^2+1)|+C