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

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

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integral of sqrt(x)arctan(sqrt(x))

\int x arctan (x) d x

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

Solution steps

\int x arctan (x) d x

Apply Integration By Parts

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (x^{4} + 12 x y^{2} + 2 y^{4})

y = \frac{2 x - 9}{3 x + 7}

\int \frac{7 ^{x}}{1 - 7 ^{2 x}} d x

\frac{d}{d x} (e^{- 3})

\frac{d}{d x} ((1 - x)^{\frac{1}{2}})

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