integral substitution

Question

Evaluate indefinite integral of (sqrt(x^2-4))/(x)

thank you.

Answer 1

Let u = √(x²-4), du = x/√(x²-4) dx, x = u²+4

∫(u²/(u²+4))du = ∫1 - 4/(u²+4)) du By dividing u² by (u²+4)

= ∫du - ∫(4/(u²+4))du

∫du = u = √(x²-4)

4∫(1/(u²+4))du

Let u=2v, du = 2dv, v = u/2

4∫(1/(v²+1))dv =2tan^-1(v) = 2tan^-1(u/2) = 2tan^-1(√(x²-4)/2)

Put the two parts together

∫(√(x²-4)/x)dx = √(x²-4)- 2tan^-1(√(x²-4)/2) + C

Answer 2

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