Evaluate ∫√(x^2-1)/x dx (upper limit is 2, lower limit is 1), using trigonometric substitution.

Question

I got tanΘ-Θ +C
I don't know how to convert this back in terms of x / change the upper and lower limits in terms of Θ.

Answer 1

draw a right triangle. If x = secΘ then tanΘ = √(x^2-1)
So your answer is
√(x^2-1) - arctan(√(x^2-1)) + C
Evaluating at the limits, you have
(√3-arctan√3)-(0-0) = √3 - π/3

If you want to change the limits, then since x = secΘ
∫[1..2] becomes ∫[0..π/3] and you can see the results will be as above.