1 / | | 1 | ----------- dx | ________ | / 2 | \/ x - 5 | / 0
Integral(1/(sqrt(x^2 - 5)), (x, 0, 1))
TrigSubstitutionRule(theta=_theta, func=sqrt(5)*sec(_theta), rewritten=sec(_theta), substep=RewriteRule(rewritten=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=tan(_theta) + sec(_theta), constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta)], context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta), context=sec(_theta), symbol=_theta), restriction=(x < sqrt(5)) & (x > -sqrt(5)), context=1/(sqrt(x**2 - 5)), symbol=x)
Now simplify:
Add the constant of integration:
The answer is:
/ | // / _________\ \ | 1 || | ___ ___ / 2 | | | ----------- dx = C + |< |x*\/ 5 \/ 5 *\/ -5 + x | / ___ ___\| | ________ ||log|------- + ------------------| for And\x > -\/ 5 , x < \/ 5 /| | / 2 \\ \ 5 5 / / | \/ x - 5 | /
/ ___\ pi*I |\/ 5 | - ---- + acosh|-----| 2 \ 5 /
=
/ ___\ pi*I |\/ 5 | - ---- + acosh|-----| 2 \ 5 /
-pi*i/2 + acosh(sqrt(5)/5)
Use the examples entering the upper and lower limits of integration.