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

The integral of x^2arccot(2x) is 1/3 x^3arccot(2x)+1/48 (4x^2+1-ln|4x^2+1|)+C

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integral of x^2arccot(2x)

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

\frac{1}{3} x^{3} arccot (2 x) + \frac{1}{48} (4 x^{2} + 1 - ln 4 x^{2} + 1) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{2 x ^{3}}{3 ( 4 x ^{2} + 1 )} d x = - \frac{1}{48} (4 x^{2} + 1 - ln 4 x^{2} + 1)

= \frac{1}{3} x^{3} arccot (2 x) - (- \frac{1}{48} (4 x^{2} + 1 - ln 4 x^{2} + 1))

Simplify

= \frac{1}{3} x^{3} arccot (2 x) + \frac{1}{48} (4 x^{2} + 1 - ln 4 x^{2} + 1)

Add a constant to the solution

= \frac{1}{3} x^{3} arccot (2 x) + \frac{1}{48} (4 x^{2} + 1 - ln 4 x^{2} + 1) + C

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What is the integral of x^2arccot(2x) ?
The integral of x^2arccot(2x) is 1/3 x^3arccot(2x)+1/48 (4x^2+1-ln|4x^2+1|)+C