First, rationalize the denominator:
color(white)=int 1/(sqrt(x+1)-sqrtx)dx=∫1√x+1−√xdx
=int 1/(sqrt(x+1)-sqrtx)color(red)(*(sqrt(x+1)+sqrtx)/(sqrt(x+1)+sqrtx))dx=∫1√x+1−√x⋅√x+1+√x√x+1+√xdx
=int (sqrt(x+1)+sqrtx)/((sqrt(x+1)-sqrtx)(sqrt(x+1)+sqrtx))dx=∫√x+1+√x(√x+1−√x)(√x+1+√x)dx
=int (sqrt(x+1)+sqrtx)/((sqrt(x+1))^2-(sqrtx)^2)dx=∫√x+1+√x(√x+1)2−(√x)2dx
=int (sqrt(x+1)+sqrtx)/(x+1-x)dx=∫√x+1+√xx+1−xdx
=int (sqrt(x+1)+sqrtx)/(color(red)cancelcolor(black)x+1color(red)cancelcolor(black)(color(black)-x))dx
=int (sqrt(x+1)+sqrtx)/1dx
=int (sqrt(x+1)+sqrtx) dx
=intsqrt(x+1) dx+intsqrtx dx
=int(x+1)^(1/2) dx+intx^(1/2) dx
Power rule:
=((x+1)^(1/2+1))/(1/2+1)+(x^(1/2+1))/(1/2+1)
=((x+1)^(3/2))/(3/2)+(x^(3/2))/(3/2)
=2/3(x+1)^(3/2)+2/3x^(3/2)
You can factor out the 2/3, and don't forget to add C:
=2/3((x+1)^(3/2)+x^(3/2))+C
That's the whole integral. Hope this helped!