1 / | | 1 | --------- dx | 2 | / 2\ | \1 + x / | / 0
Integral(1/((1 + x^2)^2), (x, 0, 1))
TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), restriction=True, context=1/((x**2 + 1)**2), symbol=x)
Now simplify:
Add the constant of integration:
The answer is:
/ | | 1 atan(x) x | --------- dx = C + ------- + ---------- | 2 2 / 2\ | / 2\ 2*\1 + x / | \1 + x / | /
1 pi - + -- 4 8
=
1 pi - + -- 4 8
1/4 + pi/8
Use the examples entering the upper and lower limits of integration.