What is the integral of 2xarctan^2(x) ?

The integral of 2xarctan^2(x) is x^2arctan^2(x)-2xarctan(x)+arctan^2(x)+2ln|sqrt(x^2+1)|+C

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integral of 2xarctan^2(x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \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

= 2 (\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}

= 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

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

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

Add a constant to the solution

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

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What is the integral of 2xarctan^2(x) ?
The integral of 2xarctan^2(x) is x^2arctan^2(x)-2xarctan(x)+arctan^2(x)+2ln|sqrt(x^2+1)|+C