Evaluate the Integral

Problem

(∫_^)(1/(x√(,9*x2−1))*d(x))

Solution

1. Identify the form of the integral as a standard inverse trigonometric substitution, specifically matching the form (∫_^)(1/(u√(,u2−a2))*d(u))=1/a*sec(|u|/a)(−1)+C

2. Rewrite the expression inside the square root to identify u and a by expressing 9*x2 as (3*x)2

(∫_^)(1/(x√(,(3*x)2−1))*d(x))

3. Perform a substitution by letting u=3*x which implies d(u)=3*d(x) and x=u/3

(∫_^)(1/(u/3√(,u2−1))d(u)/3)

4. Simplify the constant factors by canceling the 3 in the denominator of x with the 3 from d(u)

(∫_^)(1/(u√(,u2−1))*d(u))

5. Apply the integration rule for the inverse secant function.

sec(|u|)(−1)+C

6. Substitute back u=3*x to obtain the final result in terms of x

sec(|3*x|)(−1)+C

Final Answer

(∫_^)(1/(x√(,9*x2−1))*d(x))=sec(|3*x|)(−1)+C

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