What is the integral of (arctan(x))/(xsqrt(x)) ?

The integral of (arctan(x))/(xsqrt(x)) is

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integral of (arctan(x))/(xsqrt(x))

\int \frac{arctan ( x )}{x x} d x

2 (- \frac{arctan ( x )}{x} + 2 (\frac{1}{4 2} (ln 2 x + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x - 22 x + 2 + 2 arctan (2 x - 1)))) + C

Solution steps

\int \frac{arctan ( x )}{x x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

: - \frac{arctan ( u ^{2} )}{u} - \int - \frac{2}{u ^{4} + 1} d u

= 2 (- \frac{arctan ( u ^{2} )}{u} - \int - \frac{2}{u ^{4} + 1} d u)

\int - \frac{2}{u ^{4} + 1} d u = - 2 (\frac{1}{4 2} (ln 2 u^{2} + 22 u + 2 + 2 arctan (2 u + 1)) + \frac{1}{4 2} (- ln 2 u^{2} - 22 u + 2 + 2 arctan (2 u - 1)))

= 2 (- \frac{arctan ( u ^{2} )}{u} - (- 2 (\frac{1}{4 2} (ln 2 u^{2} + 22 u + 2 + 2 arctan (2 u + 1)) + \frac{1}{4 2} (- ln 2 u^{2} - 22 u + 2 + 2 arctan (2 u - 1)))))

Substitute back

u = x

= 2 - \frac{arctan ( ( x ) ^{2} )}{x} - (- 2 (\frac{1}{4 2} (ln 2 (x)^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 (x)^{2} - 22 x + 2 + 2 arctan (2 x - 1))))

Simplify

2 - \frac{arctan ( ( x ) ^{2} )}{x} - (- 2 (\frac{1}{4 2} (ln 2 (x)^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 (x)^{2} - 22 x + 2 + 2 arctan (2 x - 1)))) : 2 (- \frac{arctan ( x )}{x} + 2 (\frac{1}{4 2} (ln 2 x + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x - 22 x + 2 + 2 arctan (2 x - 1))))

= 2 (- \frac{arctan ( x )}{x} + 2 (\frac{1}{4 2} (ln 2 x + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x - 22 x + 2 + 2 arctan (2 x - 1))))

Add a constant to the solution

= 2 (- \frac{arctan ( x )}{x} + 2 (\frac{1}{4 2} (ln 2 x + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x - 22 x + 2 + 2 arctan (2 x - 1)))) + C

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What is the integral of (arctan(x))/(xsqrt(x)) ?
The integral of (arctan(x))/(xsqrt(x)) is