How can i evaluate the integral : Pi/2 $(1-3) Sqrt (t^2-1)dt

[Riazy]

Hi how can i evaluate this integral? ( see scanned paper)

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[galactus]

Re: How can i evaluate the integral : Pi/2 $(1-3) Sqrt (t^2-

Trig sub is commonly used for this integral or its brethren.

But, I prefer the hyperbolic trig sub with integrals of the form $\displaystyle \sqrt{t^{2}-a^{2}}$

$\displaystyle \int\sqrt{t^{2}-1}dt$

Let $\displaystyle t=cosh(u), \;\ dt=sinh(u)du$

Because $\displaystyle cosh^{2}(u)-1=sinh^{2}(u)$

this leads to:

$\displaystyle \int \sqrt{cosh^{2}(u)-1}sinh(u)du=\int sinh^{2}(u)du$

$\displaystyle \frac{1}{2}\int (cosh(2u)-1)du$

Now, it is easier to finish.