The value of the integral∫−11 ddxtan−1⁡1xdx is

ddxtan−1⁡1x=11+1x2−1x2=−11+x2⇒ I=∫−11 ddxtan−1⁡1xdx=∫−11 −dx1+x2=−tan−1⁡x−11=−π4+−π4=−π2Note that I=tan−1⁡(1/x)−11=π4−−π4=π2 is incorrect,since the function tan−1⁡(1/x) is not an antiderivative of (d/dx)tan−1⁡(1/x) on the interval[−1,1]