In the exercise we want to use a substitution to integrate integral xsqrt(x+1)dx.

Question

Find the substitution u to transform the integral into integral(u-1)u^(1/2)du. u=?

Answer 1

If you take u = x + 1

Then, x + 1 = u => sqrt(x+1) = sqrt(u) = u^(1/2)

Then, x = (u - 1)

And, du/dx = 1 => du = dx

So,

x*sqrt(x+1)*dx
Becomes
(u-1)*sqrt(u)*du = (u-1)u^(1/2)du

With the substitution u = x + 1