Explained: "Evaluate the following definite integrals. ∫535dxx2−9"

aortiH

Answered question

2021-02-25

Evaluate the following definite integrals.

\int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2} - 9}}

au4gsf

Skilled2021-02-26Added 95 answers

Given integrals is \int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2} - 9}}

Consider x=3 sec(u)

Suvstitute x=3 sec(u) in \int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2} - 9}}

\int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2} - 9}} = \int_{\sec^{- 1} (\frac{5}{4})}^{\sec^{- 1} (\sqrt{5})} \sec (u) d u

= [\ln (\tan u + \sec u)]_{\sec^{- 1} (\frac{5}{4})}^{\sec^{- 1} (\sqrt{5})}

= \ln (\tan (\sec^{- 1} (\sqrt{5})) + \sec (\sec^{- 1} (\sqrt{5}))) - \ln (\tan (\sec^{- 1} (\frac{5}{3})) + \sec (\sec^{- 1} (\frac{5}{3})))

= \ln (2 + \sqrt{5}) - \ln 3

= 0.34502

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