Evaluate the Integral

Problem

(∫_^)(1/√(,x2−9)*d(x))

Solution

1. Identify the form of the integral as a trigonometric substitution problem involving √(,x2−a2) where a=3

2. Substitute x=3*sec(θ) which implies d(x)=3*sec(θ)*tan(θ)*d(θ)

3. Simplify the radical using the identity sec2(θ)−1=tan2(θ) resulting in √(,9*sec2(θ)−9)=3*tan(θ)

4. Rewrite the integral in terms of θ

(∫_^)((3*sec(θ)*tan(θ))/(3*tan(θ))*d(θ))

5. Simplify the integrand to obtain:

(∫_^)(sec(θ)*d(θ))

6. Integrate using the standard formula for the integral of the secant function:

ln(sec(θ)+tan(θ))+C

7. Back-substitute to return to the variable x using sec(θ)=x/3 and tan(θ)=√(,sec2(θ)−1)=√(,x2−9)/3

ln(x/3+√(,x2−9)/3)+C

8. Simplify the logarithmic expression by combining the fractions and absorbing the constant −ln(3) into the constant of integration C

ln(x+√(,x2−9))+C

Final Answer

(∫_^)(1/√(,x2−9)*d(x))=ln(x+√(,x2−9))+C

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