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

The integral of rctan(sqrt(x)) is dxarctan(sqrt(x))+darctan(sqrt(x))-dsqrt(x)+C

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integral of rctan(sqrt(x))

\int d arctan (x) d x

d x arctan (x) + d arctan (x) - d x + 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 u-substitution

: \int arctan (u) \cdot 2 u d u

= d \cdot \int arctan (u) \cdot 2 u d u

Take the constant out:

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

= d \cdot 2 \cdot \int arctan (u) u d u

Apply Integration By Parts

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

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

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

= d \cdot 2 (\frac{1}{2} u^{2} arctan (u) - \frac{1}{2} (- arctan (u) + u))

Substitute back

u = x

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

Simplify

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

= d x arctan (x) + d arctan (x) - d x

Add a constant to the solution

= d x arctan (x) + d arctan (x) - d x + C

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What is the integral of rctan(sqrt(x)) ?
The integral of rctan(sqrt(x)) is dxarctan(sqrt(x))+darctan(sqrt(x))-dsqrt(x)+C