Make a substitution to express the integrand as a rational function and then evaluate the integral.?

Question

Given: The integral of 1 / [x * sqrt.(x+1)] dx.
The answer is ln | (sqrt. (x+1) - 1 ) / (sqrt.(x+1) + 1) | + C.

Answer 1

Let x + 1 = y^2; then dx = 2y dy, and x sqrt (x + 1) = (y^2 -1) y.

So: dx/(x sqrt[x + 1]) becomes 2y dy/[(y^2 - 1) y] = 2 dy/(y^2 - 1).

But 2/(y^2 - 1) = 1/(y - 1) - 1/(y + 1). The integral of this with respect to y is:

ln |(y - 1)| - ln |(y + 1)| + C, or ln |(y - 1) /(y + 1)| + C,

where y = sqrt (x + 1); this is the answer you quoted.

Live long and prosper.

P.S. The next responder, a fellow countryman (?), was a little bit too swift. His "2" ended up in the denominator; it should have been in the numerator. It's because of one of those "Tower of Babel" fractions: 1/1/2 --- that's 2, not 1/2. I generally find that defining the connection between the variables as I did, above, eliminates the 1/2's (from differentiation) in the rest of such calculations. I've learned that I, at least, make fewer mistakes that way.

Answer 2

substitute y = sqrt(x+1) so dy/dx = 1/(2*sqrt(x+1)) and x = y^2-1

then int(1/(x*sqrt(x+1))dx) = int(1/(2*(y^2-1)))dy) then use partial fractions :-)