explain how to evaluate a square root integrand give an exam

explain how to evaluate a square root integrand

give an example

Solution

Int f(x) dx =Int dx/[1+sqrt(x-1)]

We\'ll substitute sqrt(x - 1) = t

We\'ll raise to square both sides:

x - 1 = t^2

x = t^2 + 1

We\'ll differentiate both sides:

dx = 2tdt

We\'ll re-write the integral in t:

Int 2tdt/(1+t) = 2Int tdt/(1+t)

We\'ll add and subtract 1 to numerator of the ratio:

2Int (t+1 - 1)dt/(1+t) = 2Int (t+1)dt/(1+t) - 2Int dt/(t+1)

We\'ll simplify and we\'ll get:

Int f(x) dx = 2Int dt - 2ln |t + 1|

We\'ll change the variable t:

Int f(x) dx = 2*sqrt(x - 1) - 2ln [sqrt(x - 1) + 1] + C