Solve the trigonometric integral int(arctan(1/(x^(1/2))))dx

 Exercise

$\int\left(arctan\left(\frac{1}{\sqrt{x}}\right)\right)dx$

 

Apply integration by parts: $u=arctan(\frac{1}{\sqrt{x}})$, $v'=1$

$x\arctan\left(\frac{1}{\sqrt{x}}\right)-\int\frac{\frac{1}{\sqrt{x}}}{1+\left(\frac{1}{\sqrt{x}}\right)^2}dx$

 

Simplify the expression

$x\arctan\left(\frac{1}{\sqrt{x}}\right)-\int\frac{\sqrt{x}}{1+x}dx$

 

The integral $-\int\frac{\sqrt{x}}{1+x}dx$ results in: $-2\sqrt{x}+2\arctan\left(\sqrt{x}\right)$

$-2\sqrt{x}+2\arctan\left(\sqrt{x}\right)$

 

Gather the results of all integrals

$x\arctan\left(\frac{1}{\sqrt{x}}\right)+2\arctan\left(\sqrt{x}\right)-2\sqrt{x}$

 

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$x\arctan\left(\frac{1}{\sqrt{x}}\right)+2\arctan\left(\sqrt{x}\right)-2\sqrt{x}+C_0$

$x\arctan\left(\frac{1}{\sqrt{x}}\right)+2\arctan\left(\sqrt{x}\right)-2\sqrt{x}+C_0$

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