What is the integral of rctan(x) ?

The integral of rctan(x) is d(xarctan(x)-1/2 ln|x^2+1|)+C

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integral of rctan(x)

\int d arctan (x) d x

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

Solution steps

\int d arctan (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= d \cdot \int arctan (x) d x

Apply Integration By Parts

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

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

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

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

Add a constant to the solution

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

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

\int e^{x} 33 + e^{x} d x

\int \frac{2 x + 9}{x ^{2} + 9 x + 3} d x

\int \frac{16 - x ^{2}}{x} d x

What is the integral of rctan(x) ?
The integral of rctan(x) is d(xarctan(x)-1/2 ln|x^2+1|)+C