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

The integral of (arctan(sqrt(x)))/(x^2) is -(arctan(sqrt(x)))/x-1/(sqrt(x))-arctan(sqrt(x))+C

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

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

- \frac{arctan ( x )}{x} - \frac{1}{x} - arctan (x) + C

Solution steps

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

Apply u-substitution

: \int \frac{2 arctan ( u )}{u ^{3}} d u

= \int \frac{2 arctan ( u )}{u ^{3}} 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 )}{u ^{3}} d u

Apply Integration By Parts

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

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

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

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

Substitute back

u = x

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

Simplify

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

= - \frac{arctan ( x )}{x} - \frac{1}{x} - arctan (x)

Add a constant to the solution

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

x = 4 y^{2}, x + y = 3

\int \frac{10 x}{81 x ^{4} + 49} d x

\int arccos (x) d x

\int_{- 2}^{2} (x^{3} - 1) d x

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