Check the solution to "Integrate ∫cosh−1(0)cosh−1(3)cosh2(t)−coshtdt"

kadetskihykw

Answered question

2022-04-21

Integrate

\int_{{cosh}^{- 1} (0)}^{{cosh}^{- 1} (3)} \sqrt{{cosh}^{2} (t) - cosh t} d t

Cynthia Herrera

Beginner2022-04-22Added 16 answers

One method is to make the substitution

u = \frac{t^{2}}{1 - t^{2}} \Rightarrow t = \sqrt{\frac{u}{u + 1}}

However, this approach needs a bit of a struggle to complete.
A more clever approach (inspired by the answer given by Wolfram|Alpha) is to substitute

u = {sinh}^{2}

Then.

\sqrt{\frac{u}{u + 1}} = \sqrt{\frac{{sinh}^{2} t}{{cosh}^{2} t}} = \frac{sinh t}{cosh t}

and

d u = 2 sinh t cosh t

Therefore, the integrand becomes:

\int_{0}^{{sinh}^{- 1} (\sqrt{3})} 2 {sinh}^{2} t d t = \frac{1}{2} \int_{0}^{{sinh}^{- 1} (\sqrt{3})} {(e^{t} - e^{- t})}^{2} d t = \frac{1}{2} \int_{0}^{{sinh}^{- 1} (\sqrt{3})} (e^{2 t} + e^{- 2 t} - 2) d t

which can be easily evaluated.

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