What is the integral of x^3arccot(x^2) ?

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

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integral of x^3arccot(x^2)

\int x^{3} arccot (x^{2}) d x

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

Solution steps

\int x^{3} arccot (x^{2}) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} arccot (x^{2}) - \int - \frac{x ^{5}}{2 ( x ^{4} + 1 )} d x

= \frac{1}{4} x^{4} arccot (x^{2}) - \int - \frac{x ^{5}}{2 ( x ^{4} + 1 )} d x

\int - \frac{x ^{5}}{2 ( x ^{4} + 1 )} d x = - \frac{1}{2} (\frac{x ^{2}}{2} - \frac{1}{2} arctan (x^{2}))

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

Simplify

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

Add a constant to the solution

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

\int (3 x)^{- 1} d x

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\int sec^{2} (t) tan^{2} (t) d t

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