How do you find the integral of #int 1/(sqrt(x)*(1+x)) dx#?

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Answer 1 · 2018-07-09T03:49:41

#int1/(sqrtx(1+x))dx=2arc tan(sqrtx)+c#

We know that,

#color(red)((1)d/(dx)(arc tanx)=1/(1+x^2)=>int1/(1+x^2)dx=arc tanx+c#

We have ,

#I=int1/(sqrtx(1+x))dx#

Subst. #color(blue)(sqrtx=u=>x=u^2=>dx=2udu#

So ,

#I=int1/(u(1+u^2))2u*du#

#=2int1/(1+u^2)du...tocolor(red)(Apply(1)#

#=2arc tan u+c#

Subst. back #color(blue)(u=sqrtx# ,we get

#I=2arc tan(sqrtx)+c#