Evaluate the integrals: a.) ∫ dx / (√x)(√ (1-x)) b.) ∫ dx / √(x^2 - 4x +3)

Question

We are supposed to make the substitution, u = √x

Answer 1

a.)I = ∫ dx / (√x)(√ (1-x)) = ∫dx/√(x - x²) = ∫dx/√[1/4 - (x - 1/2)²]; substitution: x - 1/2 = 1/2sint; dx = 1/2costdt

I = ∫1/2costdt/1/2cost = ∫dt = t + C = sin^-1(2x - 1) + C;

b.) I = ∫dx /√(x² - 4x + 3) = ∫dx/√[(x - 2)² - 1]

Subst: x - 2 = sect; dx = sect·tantdt; I = ∫sect·tantdt/

tant = ∫sectdt = lnIsect + tantI + C = lnIx - 2 + √(x² - 4x + 3)I + C

Answer 2

a)

integral of dx/sqr(x)sqr(1-x)

= ∫dx/√(x-x^2)

= arcsin(2x-1) + C

try looking it up in some integral tables, in some calculus books it's in the appendix or back or front hard covers pages, and using u substitutions

b)

integral of dx/square root of (x^2 -4x +3)

= natural log of (absolute value of x+square root of (x-2)^2-1) - 2) plus an unknown constant

= ln((|x+√[(x-2)^2-1] -2|) + C

or "simplify" further,

as what's under the radical (x-2^2) -1

= x^2-4x+4-1

= x^2 -4x+3

try integral tables

or better yet, plug the problem into an on line integral calculator, with step by step method to solve it

let u =x-2

then du= dx

∫(1/√√(u^2-1)du

= ln|(√u^2-1)+u]|+ C

= ln|√[x-2)²-1]+x-2| + C

no guarantees on the arithmetic or algebra manipulations, without errors but it's the basic calculus methods

look at the integral tables and the online integral calculators. Also try some trig substitutions.

Together you can see best what's going on.