What is the integral of (sqrt(arctan(x)))/(1+x^2) ?

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

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integral of (sqrt(arctan(x)))/(1+x^2)

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

\frac{2}{3} arctan^{\frac{3}{2}} (x) + C

Solution steps

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

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = arctan (x)

= \frac{2}{3} arctan^{\frac{3}{2}} (x)

Add a constant to the solution

= \frac{2}{3} arctan^{\frac{3}{2}} (x) + C

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