Question

find integral of:

Answer 1

\[4\int\limits_{3}^{5}\sqrt(\frac{ x ^{2} }{ 9 }-1)\]

Answer 2

\[\large4\int_3^5\sqrt{\frac{x^2}{9}-1}\;dx\;\;\;?\]

Answer 3

If yes, let \(x=3\sec u\), so \(dx=3\sec u\tan u\;du\).

\[\large4\int_{\sec^{-1}(1)}^{\sec^{-1}(5/3)}\sqrt{\frac{1}{9}(3\sec u)^2-1}\;(3\sec u\tan u\;du)\\
\large12\int_{\sec^{-1}(1)}^{\sec^{-1}(5/3)}\sqrt{\sec ^2u-1}\;\sec u\tan u\;du\\
\large12\int_{\sec^{-1}(1)}^{\sec^{-1}(5/3)}\sqrt{\tan^2u}\;\sec u\tan u\;du\\
\large12\int_{\sec^{-1}(1)}^{\sec^{-1}(5/3)}\tan^2u\sec u\;du\]
and so on.