How do you integrate int 1 / (sqrt(x+1) - sqrt(x)) ?

2 Answers
Apr 2, 2018

The integral is equal to 2/3((x+1)^(3/2)+x^(3/2))+C.

Explanation:

First, rationalize the denominator:

color(white)=int 1/(sqrt(x+1)-sqrtx)dx

=int 1/(sqrt(x+1)-sqrtx)color(red)(*(sqrt(x+1)+sqrtx)/(sqrt(x+1)+sqrtx))dx

=int (sqrt(x+1)+sqrtx)/((sqrt(x+1)-sqrtx)(sqrt(x+1)+sqrtx))dx

=int (sqrt(x+1)+sqrtx)/((sqrt(x+1))^2-(sqrtx)^2)dx

=int (sqrt(x+1)+sqrtx)/(x+1-x)dx

=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!

Apr 2, 2018

See below.

Explanation:

1 / (sqrt(x+1) - sqrt(x)) = 1 / (sqrt(x+1) - sqrt(x))((sqrt(x+1) + sqrt(x)) /(sqrt(x+1) + sqrt(x)) ) = sqrt(x+1) + sqrt(x)

hence

int\ dx / (sqrt(x+1) - sqrt(x))= int\(sqrt(x+1) + sqrt(x))dx