Evaluate the integral and express your answer in simplest form. int(1)/(xsqrt(x^(2)-25))dx Answer:

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Evaluate the integral and express your answer in simplest form.  int(1)/(xsqrt(x^(2)-25))dx Answer:

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Recognize standard inverse trigonometric form: Recognize the integral as a standard inverse trigonometric form. The integral ∫(1)/(x*sqrt(x^2-25))dx can be recognized as an inverse trigonometric integral because the denominator has the form of a^2 - u^2, where a = 5 and u = x. This suggests a substitution using the inverse trigonometric function secant.Perform substitution with secant: Perform the substitution x = 5sec(θ). Let x = 5sec(θ), which implies dx = 5sec(θ)tan(θ)dθ. We also need to express sqrt(x^2-25) in terms of θ. Since x = 5sec(θ), we have sqrt(x^2-25) = sqrt(25sec^2(θ)-25) = sqrt(25(tan^2(θ))) = 5tan(θ).Rewrite integral in terms of theta: Rewrite the integral in terms of θ. Substituting x and dx into the integral, we get ∫(1)/(5sec(θ)*5tan(θ)) * 5sec(θ)tan(θ)dθ. The 5's and sec(θ)tan(θ) cancel out, leaving us with ∫(1)/(5tan(θ)) * 5sec(θ)tan(θ)dθ = ∫dθ.Integrate with respect to theta: Integrate with respect to θ. The integral of dθ is simply θ. So we have ∫dθ = θ + C, where C is the constant of integration.Back-substitute for x: Back-substitute to express the answer in terms of x. We originally let x = 5sec(θ), so we need to solve for θ in terms of x. We have sec(θ) = x/5, and taking the inverse secant of both sides gives us θ = sec^(-1)(x/5). Therefore, the integral in terms of x is sec^(-1)(x/5) + C.

Evaluate the integral and express your answer in simplest form. $\newline$ $\int \frac{1}{x \sqrt{x^{2}-25}} d x$ $\newline$ Answer:

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Evaluate the integral and express your answer in simplest form. $\newline$ $\int \frac{1}{x \sqrt{x^{2}-25}} d x$ $\newline$ Answer: