Evaluate the following definite integrals. int_5^{3sqrt5}frac{dx}{sqrt{x^2-9}}

Evaluate the following definite integrals. int_5^{3sqrt5}frac{dx}{sqrt{x^2-9}}

Question
Integrals
asked 2021-02-25
Evaluate the following definite integrals.
\(\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}\)

Answers (1)

2021-02-26
\(\text{Given integrals is }\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}\)
\(\text{Consider x=3 sec(u)}\)
\(\text{Suvstitute x=3 sec(u) in }\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}\)
\(\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}=\int_{\sec^{-1} (\frac{5}{4})}^{\sec^{-1}(\sqrt5)} \sec(u)du\)
\(=[\ln(\tan u+\sec u)]_{\sec^{-1} (\frac{5}{4})}^{\sec^{-1}(\sqrt5)}\)
\(=\ln(\tan(\sec^{-1}(\sqrt5))+\sec(\sec^{-1}(\sqrt5)))-\ln(\tan(\sec^{-1}(\frac{5}{3}))+\sec(\sec^{-1}(\frac{5}{3})))\)
\(=\ln(2+\sqrt5)-\ln3\)
\(=0.34502\)
0

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