Evaluate the Integral

Problem

(∫_^)(1/(x√(,x2−4))*d(x))

Solution

1. Identify the form of the integral as a standard trigonometric substitution or a known inverse trigonometric derivative form.

2. Apply the substitution x=2*sec(θ) which implies d(x)=2*sec(θ)*tan(θ)*d(θ)

3. Substitute the expressions into the integral:

(∫_^)((2*sec(θ)*tan(θ))/(2*sec(θ)√(,(2*sec(θ))2−4))*d(θ))

4. Simplify the radical using the identity sec2(θ)−1=tan2(θ)

√(,4*sec2(θ)−4)=√(,4*(sec2(θ)−1))=2*tan(θ)

5. Reduce the integrand by canceling common terms:

(∫_^)((2*sec(θ)*tan(θ))/(2*sec(θ)⋅2*tan(θ))*d(θ))=(∫_^)(1/2*d(θ))

6. Integrate with respect to θ

1/2*θ+C

7. Back-substitute to return to the variable x using sec(θ)=x/2 which means θ=arcsec(x/2)

1/2*arcsec(x/2)+C

Final Answer

(∫_^)(1/(x√(,x2−4))*d(x))=1/2*arcsec(x/2)+C

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