1 / | | 1 | 1*----------- dx | ________ | / 2 | \/ x + 4 | / 0
Integral(1/sqrt(x^2 + 4), (x, 0, 1))
TrigSubstitutionRule(theta=_theta, func=2*tan(_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=True, context=1/sqrt(x**2 + 4), symbol=x)
Now simplify:
Add the constant of integration:
The answer is:
/ / ________ \ | | / 2 | | 1 | / x x| | 1*----------- dx = C + log| / 1 + -- + -| | ________ \\/ 4 2/ | / 2 | \/ x + 4 | /
asinh(1/2)
=
asinh(1/2)
Use the examples entering the upper and lower limits of integration.