Question

definite integral from 1 to 2 of sqrt((x^2) - 1)/x. I recognize that if x = sec(t), then the function becomes tan(t)/sec(t), but where do I go from there? Thanks.

Answer 1

you have to replace dx with dt

x = sec(t)
dx = sec(t) tan(t) dt
\[\int\limits \frac{\tan t}{\sec t} \sec t \tan t dt = \int\limits \tan^{2} t dt = \int\ sec^{2}t -1 = \tan t - t\]

Answer 2

Thank you so much! How do I translate the 1 to 2 into the substituted integral?

Answer 3

well
\[t = \sec^{-1}(x)\]
\[\sec^{-1}(1) = 0\]
\[\sec^{-1}(2) = \pi/3\]

Answer 4

Ah, yes. That's right. I appreciated the help.

Answer 5

yw