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

The integral of x*arctan(x) is 1/2 x^2arctan(x)-1/2 (-arctan(x)+x)+C

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integral of x*arctan(x)

\int x \cdot arctan (x) d x

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

Solution steps

\int x arctan (x) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

x (1 - \frac{4}{x + 3})

\frac{d}{d x} (\frac{2}{e ^{\frac{x}{2}} + e ^{- \frac{x}{2}}})

f (x) = 5 tan (3 x)

n = 1 \sum \infty \frac{n - 5}{n + 2}

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