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

The integral of arctan(sqrt(x)) is xarctan(sqrt(x))+arctan(sqrt(x))-sqrt(x)+C

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integral of arctan(sqrt(x))

\int arctan (x) d x

x arctan (x) + arctan (x) - x + C

Solution steps

\int arctan (x) d x

Apply u-substitution

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

= \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

= 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

= 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)

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

Substitute back

u = x

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

Simplify

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

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

Add a constant to the solution

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

\int sec^{4} (2 x) tan (2 x) d x

e^{- \frac{x}{2}}

\int 6 y^{- 4} d y

5 x - x^{2}, (4, 4)

\int 5 - 2 x d x

What is the integral of arctan(sqrt(x)) ?
The integral of arctan(sqrt(x)) is xarctan(sqrt(x))+arctan(sqrt(x))-sqrt(x)+C