Integration, where have i gone wrong?

[Dell]

\intarctg\sqrt{x}dx

using\intudv=uv-\intvdu

dv=dx
v=x

u=arctg\sqrt{x}
du=u'dx=[1/(1+x)]dx

\intarctg\sqrt{x}dx=xarctg\sqrt{x} - \intx*[1/(1+x)]dx
=xarctg\sqrt{x} - \int[x/(1+x)]dx
=xarctg\sqrt{x} - \int1-[1/(1+x)]dx
=xarctg\sqrt{x} - x+ln|x+1|+c
but this is wrong, where have i made the mistake?

 

[MathematicalPhysicist]

The derivative of arctan(sqrt(x)) is (1/(1+x))*1/2(sqrt(x)), you can still solve this question by looking at:
sqrt(x)/(1+x)dx=2/3(d(x^3/2))/(1+(x^3/2)^(2/3))
sqrt(x)dx=2/3d(x^3/2)
and the integral of dy/(1+y^2/3)
can be solved by writing y^1/3=sinh(t).

I am not sure this will work, but I think that there is no easy way here.

 

[Dell]

thanks