Evaluate the integral: int(2)/(xsqrt(x^(2)-36))dx

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Identify Integral: Let's identify the integral we need to solve: I = ∫(2)/(x√(x^2-36))dx We can see that this integral suggests a trigonometric substitution because of the form a^2 - x^2 under the square root. We will use the substitution x = asec(θ), where a = 6 in this case, because 36 = 6^2.Substitute x: Substitute x = 6sec(θ) into the integral. Then, dx = 6sec(θ)tan(θ)dθ and x^2 - 36 = 36sec^2(θ) - 36 = 36(tan^2(θ)). The integral becomes: I = ∫(2)/(6sec(θ)√(36tan^2(θ))) * 6sec(θ)tan(θ)dθSimplify Integral: Simplify the integral by canceling terms and using the identity √(tan^2(θ)) = |tan(θ)|. Since we are dealing with x > 6 (because of the original square root), sec(θ) > 0, and thus tan(θ) > 0, we can remove the absolute value: I = ∫(2)/(6sec(θ)tan(θ)) * 6sec(θ)tan(θ)dθ I = ∫(2)dθIntegrate with θ: Integrate with respect to θ: I = 2θ + C, where C is the constant of integration.Substitute back for θ: We need to substitute back for θ using our original substitution x = 6sec(θ). We have sec(θ) = x/6, and to find θ, we use the definition of secant: sec(θ) = 1/cos(θ), so cos(θ) = 6/x. Then θ = cos^(-1)(6/x). I = 2cos^(-1)(6/x) + C

Evaluate the integral: $\int \frac{2}{x \sqrt{x^{2}-36}} d x$

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Evaluate the integral: ∫2xx2−36dx \int \frac{2}{x \sqrt{x^{2}-36}} d x ∫xx2−36​2​dx

Evaluate the integral: $\int \frac{2}{x \sqrt{x^{2}-36}} d x$