Evaluate the Integral

Problem

(∫_^)(√(,x^2−4)/x*d(x))

Solution

1. Identify the form of the integrand, which suggests a trigonometric substitution because of the √(,x^2−a^2) term.

2. Substitute x=2*sec(θ) which implies d(x)=2*sec(θ)*tan(θ)*d(θ)

3. Simplify the radical using the identity √(,(2*sec(θ))^2−4)=√(,4*(sec^2(θ)−1))=√(,4*tan^2(θ))=2*tan(θ)

4. Rewrite the integral in terms of θ

(∫_^)((2*tan(θ))/(2*sec(θ))⋅2*sec(θ)*tan(θ)*d(θ))

5. Simplify the expression by canceling 2*sec(θ)

(∫_^)(2*tan^2(θ)*d(θ))

6. Apply the identity tan^2(θ)=sec^2(θ)−1

2*(∫_^)((sec^2(θ)−1)*d(θ))

7. Integrate with respect to θ

2*(tan(θ)−θ)+C

8. Back-substitute to return to x Since sec(θ)=x/2 then θ=arcsec(x/2) and tan(θ)=√(,x^2−4)/2

9. Distribute the constant and simplify:

2*(√(,x^2−4)/2−arcsec(x/2))+C

√(,x^2−4)−2*arcsec(x/2)+C

Final Answer

(∫_^)(√(,x^2−4)/x*d(x))=√(,x^2−4)−2*arcsec(x/2)+C

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