What is the integral of sqrt(x)arctan(1/(sqrt(x))) ?

The integral of sqrt(x)arctan(1/(sqrt(x))) is 2/3 x^{3/2}arctan(1/(sqrt(x)))+1/3 (x+1-ln|x+1|)+C

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integral of sqrt(x)arctan(1/(sqrt(x)))

\int x arctan (\frac{1}{x}) d x

\frac{2}{3} x^{\frac{3}{2}} arctan (\frac{1}{x}) + \frac{1}{3} (x + 1 - ln ∣ x + 1 ∣) + C

Solution steps

\int x arctan (\frac{1}{x}) d x

Apply Integration By Parts

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

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

\int - \frac{x}{3 ( x + 1 )} d x = - \frac{1}{3} (x + 1 - ln ∣ x + 1 ∣)

= \frac{2}{3} x^{\frac{3}{2}} arctan (\frac{1}{x}) - (- \frac{1}{3} (x + 1 - ln ∣ x + 1 ∣))

Simplify

= \frac{2}{3} x^{\frac{3}{2}} arctan (\frac{1}{x}) + \frac{1}{3} (x + 1 - ln ∣ x + 1 ∣)

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} arctan (\frac{1}{x}) + \frac{1}{3} (x + 1 - ln ∣ x + 1 ∣) + C

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What is the integral of sqrt(x)arctan(1/(sqrt(x))) ?
The integral of sqrt(x)arctan(1/(sqrt(x))) is 2/3 x^{3/2}arctan(1/(sqrt(x)))+1/3 (x+1-ln|x+1|)+C